The altitude (or height) of a triangle is increasing at a rate of 2.5cm/min while the area of the triangle is increasing at a rate of 3cm2/min. At what rate (in cm/min) is the base of the triangle changing when the altitude is 12cm and the area is 84cm2 Round your answer to three decimal places.

Answers

Answer 1

The base is decreasing at 1.917 cm/min when altitude is 12cm and area=84cm².

Let A be the area of the triangle, h be the height of the triangle, and b be the base of the triangle.

Then the formula for the area of a triangle is:

A = (1/2)bh

We are given that dh/dt = 2.5 cm/min (the height is increasing at a rate of 2.5cm/min), and dA/dt = 3 cm²/min (the area is increasing at a rate of 3cm²/min).

We want to find db/dt, the rate of change of the base of the triangle when h = 12 cm and A = 84 cm².

To solve this problem, we need to use the chain rule of differentiation.

We start by differentiating both sides of the formula for the area of a triangle with respect to time t:

dA/dt = (1/2) d/dt (bh)

Next, we can use the product rule of differentiation to find d/dt (bh):

d/dt (bh) = b dh/dt + h db/dt

Substituting this into the previous equation gives:

dA/dt = (1/2) [ b dh/dt + h db/dt ]

Now we can substitute the given values of dh/dt and dA/dt, as well as h = 12 cm and A = 84 cm².

To find db/dt:

3 cm²/min = (1/2) [ b (2.5 cm/min) + 12 cm db/dt ]

Simplifying this expression gives:

6 cm²/min = 2.5 b cm²/min + 12 cm db/dt

Substituting A = 84 cm² and h = 12 cm into the formula for the area of a triangle gives:

84 cm² = (1/2) b (12 cm)

Simplifying this expression gives:

b = 14 cm

Now we can substitute b = 14 cm into the previous equation to find db/dt:

6 cm²/min = 2.5 (14 cm) cm²/min + 12 cm db/dt

Simplifying this expression gives:

db/dt = (6 cm²/min - 35 cm²/min) / (12 cm)

db/dt = -1.917 cm/min (rounded to three decimal places)

Therefore, the base of the triangle is decreasing at a rate of 1.917 cm/min when the height is 12 cm and the area is 84 cm².

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Related Questions

LaShawn designs websites for local businesses. He charges $25 an hour to build a website, and charges $15 an hour to update websites once he builds them. He wants to earn at least $100 every week, but he does not want to work more than 6 hours each week. What is a possible weekly number of hours LaShawn can spend building websites x and updating websites y that will allow him to obtain his goals?

Answers

Answer:

1 hour to build a website

5 hours to update websites

Step-by-step explanation:

x is the hours to build a website

y is the hours to update websites

x + y = 6 ------> 25x + 25y = 150

25x + 15y = 100

10y = 50

y = 5

25x + 15 (5) = 100

25x + 75 = 100

25x = 25

x = 1

So, LaShawn needs 1 hour to build a website and 5 hours to update websites to allow him to reach his goals.

A freezer chest is in the shape of a rectangular prism. Measured on the inside, the chest is 4 feet wide, 2. 5 feet tall, and 2 feet long. How much space is inside to hold frozen foods?

Answers

The freezer chest has 20 cubic feet of space inside to hold frozen foods.

To find the amount of space inside the freezer chest, we need to calculate its volume. The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height.

Using the measurements given, we can plug them into the formula and calculate:

V = 4 ft x 2.5 ft x 2 ft
V = 20 cubic feet

Therefore, there is 20 cubic feet of space inside the freezer chest to hold frozen foods.

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The base of a solid is the region in the first quadrant between the graph of y=2x
and the x-axis for 0≤x≤1. For the solid, each cross section perpendicular to the x-axis is a quarter circle with the corresponding circle’s center on the x-axis and one radius in the xy-plane. What is the volume of the solid?

Answers

The volume of the solid is A. [tex]\pi/3[/tex]

What is the volume of a solid?

The volume of a solid in geometry signifies the space it occupies inside a three-dimensional area. It denotes how much content can fill up its inner region, and as usual, measured in units like cubic feet, meters, or centimeters.

Finding the measurement formula varies from shape to shape but commonly involves multiplying width, height, and length. Given that many sectors rely on this term, such as engineering, architecture, and physics, understanding the concept of the volume of a solid weighs significantly.

If we take a cross-section of (x,y)perpendicular to the x-axis, with width dx, now the cross-section is a quarter circle with radius y.

Thus, the volume of the cross-section, since y = 2x becomes: [tex]\pi x^2 dx[/tex]

Now, the volume of the solid when integrated becomes: [tex]\frac{\pi }{3}[/tex]

Option A is correct.

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Almost all employees working for financial companies in New York City receive large bonuses at the end of the year. A sample of employees selected from financial companies in New York City showed that they received an average bonus of last year with a standard deviation of. Construct a confidence interval for the average bonus that all employees working for financial companies in New York City received last year.

Round your answers to cents.

$________ to _______ $

Answers

To construct a confidence interval for the average bonus that all employees working for financial companies in New York City received last year, we need to know the sample size and the level of confidence. Without this information, we cannot calculate the confidence interval. Please provide the sample size and the level of confidence to proceed with the solution.

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A rectangular pyramid has a volume of 480 in. If a rectangular prism has a base and height congruent to the pyramid, what
is the volume of the prism? († point)

Please help!!!

Answers

After considering all the given data we come to the conclusion that the volume of the prism is 1440 in³, under the condition that A rectangular pyramid has a volume of 480 in.

The volume of a rectangular pyramid is represented by the formula
(1/3) × base area × height.
Now, the volume of a rectangular prism is given by the formula
base area × height.
Now if we consider the rectangular prism has a base and height congruent to the pyramid, then the base area of the prism is equivalent to the base area of the pyramid. Then, the volume of the prism is equivalent to three times that of the pyramid.

Hence, the volume of the pyramid is 480 in³, we can evaluate the volume of the prism
Volume of prism = 3 × Volume of pyramid
= 3 × (1/3) × Base area × Height
= Base area × Height
Then, the volume of the rectangular prism is
480 × 3
= 1440 in³.
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What relationship do you notice between the amount Tim has saves and the amount Jill has saved each week?

Answers

a. The Taylors may want to avail themselves of the help of a professional investment advisor.

b. They may prefer to find a reputable planner with appropriate credentials and experience

c,. The Taylors should track their expenses more closely because overspending without replacement income can be disastrous

How can this portfolio be done?

Because successfully managing a large investment portfolio takes a great deal of time and knowledge, the Taylors may want to avail themselves of the help of a professional investment advisor.

2) They may prefer to find a reputable planner with appropriate credentials and experience. It will be important for them to shop around to find someone with whom they feel comfortable. A fee-only planner might be the best choice, especially if their current investments are doing well and the Taylors are not interested in making big changes that would generate sales, and commissions, for the planner

e) Whether or not Tim and Jill continue to work with a financial planner depends on their financial knowledge, time and commitment. Given their successful, independent, management of their financial situation to date, they may want to develop their own plan and have it reviewed by a planner as confirmation that they are on the right track.

f) The Taylors should track their expenses more closely because overspending without replacement income can be disastrous. In the event of an unexpectedly bad financial situation or a long downturn in the economy, they would not have the time or resources to rectify their misfortune and achieve their goals.

Their big five expenses are likely to be the same as the average U.S. household - taxes, food, housing, medical care and transportation. Most retirement benefits will be taxable, as will other investment earnings. Depending on the age of the house or appliances, repairs or replacements may be necessary.

Answers

The total area of the figure is 215π square feet

Calculating the total area of the figure

From the question, we have the following parameters that can be used in our computation:

Radius, r = 5 cm

Height, h = 14 cm

So, we have

A1 = πr²

A1 = π * 5² = 25π

A2 = 2πrh

A2 = 2 * π * 5 * 14 = 140π

A3 = 1/2(4πr²)

A3 = 1/2(4π * 5²) = 50π

So, we have

Total area = 25π + 140π + 50π

Evaluate

Total area = 215π

Hence, the total area is 215π

Answers

No, the values in the table do not represent a proportional relationship because the y-values are not a constant multiple of the corresponding x-values.

Based on the given table:

x | 0 | 1 | 2 | 3
y | 0 | 3 | 5 | 6

The values in the table do not represent a proportional relationship. To be proportional, all of the y-values should be a constant multiple of the corresponding x-values. However, in this case, the ratio of y/x is not constant (3/1, 5/2, 6/3). Therefore, the relationship is not proportional.

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I need help with this one

solve for x

Answers

Answer:

x = 2

Step-by-step explanation:

A secant is a straight line that intersects a circle at two points.

A segment is part of a line that connects two points.

According to the Intersecting Secants Theorem, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

The given diagram shows two secant segments that intersect at an exterior point.

One secant segment is (6x - 1 + 7) and its external part is 7.The other secant segment is (x + 3 + 9) and its external part is 9.

Therefore, according to the Intersecting Secants Theorem:

[tex](6x-1+7) \cdot 7=(x+3+9) \cdot 9[/tex]

Solve for x:

[tex]\begin{aligned}(6x+6) \cdot 7&=(x+12) \cdot 9 \\42x+42&=9x+108\\42x+42-9x&=9x+108-9x\\33x+42&=108\\33x+42-42&=108-42\\33x&=66\\33x\div33&=66\div33\\x&=2 \end{aligned}[/tex]

Therefore, the value of x is x = 2.

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If TQ=8, what is the circumference of the circle?

Answers

The circumference of the given circle with TQ = 8 units is given by approximately 50.26 units.

We know that the formula for the circumference of a circle with radius of 'r' units is given by,

P = 2πr

Here in the given figure we can see that the length TQ is a radius for the given circle with center at point Q.

Given the value of TQ = 8 units.

So, radius = 8 units.

So the circumference of the circle is given by

= 2πr

= 2π*8

= 16π

= 50.26 units [taking π = 3.14 and approximating the value to the two decimal places]

Hence the circumference of circle is 50.26 units approximately.

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Of 100 random students surveyed, 42 own a dog, 34 own a cat, 15 own a dog and a cat, and 9 own neither a dog nor a cat. Based upon the results, how many of the next 20 students surveyed would you expect to own a dog and a cat?

Answers

In the next 20 students surveyed, you would expect 5 to own a dog and a cat

How many of the next 20 students surveyed would you expect to own a dog and a cat?

From the question, we have the following parameters that can be used in our computation:

Dog = 42

Cat = 34

Dog and cat = 15

Neither = 9

This means that

P(Dog and cat) = 15/100

When evaluated, we have

P(Dog and cat) = 5/20

So, when the next 20 students surveyed, we have

Dog and cat = 5/20 * 20

Evaluate

Dog and cat = 5

Hence, the number of dogs and cats is 5

Answers

it is crucial to ensure that data collection methods and instruments meet the required accuracy standards.

How can Digitization positional errors can still be prevented?

Digitization positional errors can still prevent accurate analysis or decision making even if they are less than the required positional accuracy of the data. This is because the accuracy of the output or results depends on the accuracy of the input data.

For example, suppose a GPS receiver is used to collect data on the location of a pipeline, and the positional accuracy requirement is 5 meters. However, due to various factors such as signal interference or poor satellite coverage, the receiver only achieves an accuracy of 2 meters. Even though the positional error is less than the required accuracy, the resulting data may still be insufficient for the intended purpose, such as accurately identifying potential hazards or planning maintenance activities.

Inaccurate data can lead to wrong decisions, increased risks, and financial losses. Therefore, it is crucial to ensure that data collection methods and instruments meet the required accuracy standards.

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Digitization positional errors may be less (say 2 meters) than the required positional accuracy of the data (say 5 meters) yet still prevent achieving the desired level of accuracy for the intended use of the data.

Ricky has 23 hours each week to dedicate to his classes. homework takes 6.5 hours and each class (c) is 1.5 hours long. how many classes does ricky take? which equation models the question? explain your thinking.

a) 23=6.5-1.5c b) 23=6.5+1.5c

c) 23=1.5+6.5c d) 23=1.5-6.5c

Answers

by dividing both sides by 1.5.

How many classes does Ricky take?

To solve the problem, we need to first determine the total amount of time Ricky spends in his classes. We know that each class is 1.5 hours long, so if he takes c classes, then he will spend a total of 1.5c hours on class time. In addition, we know that he spends 6.5 hours on homework. Therefore, the total amount of time Ricky spends on his classes and homework is:

Total time = Class time + Homework time

Total time = 1.5c + 6.5

We also know that Ricky has 23 hours per week to dedicate to his classes and homework. Therefore, we can set up the following equation:

Total time = 23

Substituting the expression for a total time from the first equation, we get:

1.5c + 6.5 = 23

Now we can solve for c:

1.5c = 23 - 6.5

1.5c = 16.5

c = 11

Therefore, Ricky takes 11 classes.

The equation that models the question is b) 23=6.5+1.5c. This equation correctly represents the total time Ricky spends on his classes and homework (23 hours), as well as the time he spends on homework (6.5 hours) and the time he spends in class (1.5c hours).

by dividing both sides by 1.5.

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3. Un negociante tiene un capital de 40 000 soles y piensa guardarlo por un periodo de 2 años. Tiene dos propuestas de bancos. Banco A: 1,5 % bimestral. Banco B: 0,5% mensual. ¿Cuál de las dos propuestas le conviene?, ¿Cuánto interés recibirá en la entidad más conveniente? (7 puntos) Alternativas

Answers

Answer:

Para comparar las propuestas de los bancos, debemos llevar las tasas de interés a una misma unidad de tiempo. Podemos convertir la tasa del Banco A de bimestral a mensual multiplicándola por 2 (ya que hay 6 bimestres en 1 año):

Tasa del Banco A: 1,5% * 2 = 3% mensual

Tasa del Banco B: 0,5% mensual

Para calcular los intereses que se obtendrán en cada banco, podemos utilizar la fórmula del interés compuesto:

I = C * ((1 + r/n)^(n*t) - 1)

Donde:

I es el interés

C es el capital inicial

r es la tasa de interés en forma decimal

n es el número de veces que se capitaliza al año

t es el tiempo en años

Para el Banco A, como la tasa está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,03/12)^(12*2) - 1) = 4 896,18 soles

Para el Banco B, como la tasa ya está en meses, capitalizaremos mensualmente (n=12):

I = 40 000 * ((1 + 0,005)^(12*2) - 1) = 4 225,48 soles

Por lo tanto, la propuesta más conveniente es la del Banco A, ya que ofrece una tasa de interés mayor y genera un interés total de 4 896,18 soles. En cambio, el Banco B genera un interés total de 4 225,48 soles.

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you are given that 4a - 2b = 10 and a + c = 3

write an expression in a,b and c that is equal to 23

give your answer in it's simplest form

Answers

Answer:

3a - 2b - c + 16 = 23

Step-by-step explanation:

so we want to write an equetion which containe a, b and c so we have given

4a - 2b = 10 and a + c = 3

so we are going to differentiate 10 to 7 + 3 it will be

4a - 2b = 10

4a - 2b = 7 + 3 ...then we insert a + c in place of 3 b/c they are equal

4a - 2b = 7 + a + c .... we take to the left side of the equal sighn

4a - a - 2b - c = 7

3a - 2b - c = 7

thrn if we want to write the equetion =23 we add 16 both side 3a - 2b - c + 16 = 23 .

Find the length of PQ.
Assume that lines which appear tangent are tangent.

Answers

Applying the secant-tangent theorem, the length of PQ is calculated as: 6 units.

How to Find the Length Using the Secant-Tangent Theorem?

In the image given, line segment QS is a secant while QP is a tangent. This, according to the secant-tangent theorem, we have:

(x - 3)(x - 3 + 5) = (x - 1)²

(x - 3)(x + 2) = (x - 1)(x - 1)

Expand:

x² - x - 6 = x² - 2x + 1

Combine like terms:

x² - x² - x + 2x = 6 + 1

x = 7

length of PQ = x - 1

Plug in the value of x:

length of PQ = 7 - 1 = 6 units.

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8: Hector’s Math test grades for the final

quarter are 89, 93, 100, 98, and 95. He has one

more test to take this quarter. All tests count

equally. What is the minimum grade Hector must

make on the last test in order to obtain an

average of at least 93?

Answers

Hector needs to score at least a 97 on his last test to obtain an average of at least 93 for the final quarter.

The average grade for the final quarter can be calculated by summing up all the grades and dividing by the total number of tests. In this case, Hector has taken 5 tests, and his grades are 89, 93, 100, 98, and 95. Therefore, his current total score is 89+93+100+98+95 = 475.

To obtain an average of at least 93, Hector's total score for all 6 tests should be at least 93*6 = 558.

So, Hector needs to score a minimum of 558 - 475 = 83 on his last test. Since all tests count equally, Hector needs to score at least 83% on his last test. Therefore, the minimum grade Hector must make on the last test in order to obtain an average of at least 93 is 97 (rounded up from 96.6).

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An aircraft leaves a town &(28n, 66°E)
and after flying 3900km due South, it
gets to another town Y calculate,
a. The radius of the line of latitude through X
the latitude of Y Correct to the nearest
degree.
(the radius of the line of latitude through Y
(Take t=2 and R=6406 km)

Answers

The radius of the line of latitude through X is approximately 5729 km, and the latitude of Y is approximately 43°.

Let X be the starting town with coordinates (28°N, 66°E), and let Y be the destination town which is 3900 km due South of X. We want to find the radius of the line of latitude through X, and the latitude of Y.

First, we can find the distance between X and Y using the Pythagorean theorem. The distance due South is 3900 km, and the distance due East is 28° × R, where R is the radius of the Earth. Using t=2 and R=6406 km, we have:

distance due East = 28° × 6406 km × cos(66°) ≈ 2055.6 km

distance between X and Y = sqrt((3900 km)^2 + (2055.6 km)^2) ≈ 4498.6 km

Next, we can find the latitude of Y using the formula:

latitude of Y = 90° - arctan(distance due South / radius of the Earth)

Using t=2 and R=6406 km, we have:

latitude of Y = 90° - arctan(3900 km / 6406 km) ≈ 43°

Finally, we can find the radius of the line of latitude through X using the formula:

radius of line of latitude = radius of the Earth * cos(latitude of X)

Using the coordinates of X, we have:

latitude of X = 28°N

radius of line of latitude = 6406 km * cos(28°) ≈ 5729 km

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Please hurry I need it ASAP

Answers

The value of x is 16.

Given,

The m line is parallel with the n line.

We need to find the value of x.

What are the relationships between parallel lines and angles?

If two lines are parallel the corresponding angles are congruent.

Example:

[tex]\sf D[/tex] /

[tex]\sf A[/tex]<----------------[tex]\sf F[/tex]/----------------------->[tex]\sf B[/tex]

[tex]\sf C[/tex] <-----------[tex]\sf M[/tex]/------------------------------>[tex]\sf D[/tex]

[tex]\sf E[/tex] /

[tex]\sf AFD = CMF[/tex]

[tex]\sf AFM = CME[/tex]

From the figure, we see that

[tex]\sf 8x + 5 + 4x - 17 = 180[/tex]

[tex]\sf 12x - 12 = 180[/tex]

[tex]\sf 12x = 180 + 12[/tex]

[tex]\sf 12x = 192[/tex]

[tex]\sf x = \dfrac{192}{12}[/tex]

[tex]\sf x = 16[/tex]

Thus the value of x is 16.

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At the performance of Seussical the Musical at your local high school, there are adult tickets and student/child tickets. You're trying to remember the cost of each to tell your music extended family to come see the musical. Your friend, her mom, and her little sister paid a total of $23 on opening night, and you know that another family paid $39 for two adults and three students. If x is cost of adult tickets and y is cost of student tickets, the two equations for these situations can be written as:

Answers

The cost of an adult ticket is $9 and the cost of a student/child ticket is $7. The solution was found by solving a system of two equations, where the variables x and y represent the costs of the two types of tickets.

Let's assign variables for the unknowns

x = cost of adult tickets

y = cost of student/child tickets

From the given information, we can create two equations

Equation 1 Friend, mom, and little sister paid a total of $23

x + 2y = 23

Equation 2 Another family paid $39 for two adults and three students

2x + 3y = 39

We now have two equations with two unknowns, which we can solve using substitution or elimination.

Here, using substitution

Solve for x in Equation 1

x = 23 - 2y

Substitute the value of x into Equation 2

2(23 - 2y) + 3y = 39

Simplify and solve for y

46 - 4y + 3y = 39

-y = -7

y = 7

Substitute the value of y into Equation 1 to solve for x

x + 2(7) = 23

x = 9

Therefore, the cost of adult tickets is $9 and the cost of student/child tickets is $7.

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Many hotel chains that offer free wi-fi service to their customers have experienced increasing demand for internet bandwidth and increasing costs. marriott international would like to test the hypothesis that the proportion of customers that are carrying two wi-fi devices exceeds 0.60. a random sample of 120 marriott customers found that 78 have two wi-fi devices. marriott international would like to set î± = 0.01. the p-value for this hypothesis test would be ________.

Answers

The p-value for this hypothesis test is approximately 0.1317.

To calculate the p-value for the hypothesis test, we need to perform a one-sample proportion test using the sample data provided.

Let's define the null and alternative hypotheses:

Null hypothesis (H₀): The proportion of Marriott customers carrying two Wi-Fi devices is equal to or less than 0.60.

Alternative hypothesis (H₁): The proportion of Marriott customers carrying two Wi-Fi devices exceeds 0.60.

Sample size (n) = 120

Number of customers with two Wi-Fi devices (x) = 78

To test the hypothesis, we can use the normal approximation to the binomial distribution since the sample size is reasonably large.

First, calculate the sample proportion:

[tex]\hat{p}[/tex] = x / n = 78 / 120 = 0.65

Next, calculate the test statistic (z-score):

z = [tex]\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n} } }[/tex]

= (0.65 - 0.60) / √((0.60 * (1 - 0.60)) / 120)

= 0.05 / √(0.24 / 120)

= 1.1180

Now, we can find the p-value corresponding to the calculated test statistic using a standard normal distribution table or a statistical calculator.

In this case, the p-value for a one-sided test (since we are testing if the proportion exceeds 0.60) is approximately 0.1317.

Therefore, the p-value for this hypothesis test is approximately 0.1317.

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Town Hall is located 4. 3 miles directly east of the middle school. The fire station is located 1. 7 miles directly north of Town Hall.

Part A

What is the length of a straight line between the school and the fire station? Round to the nearest tenth. Enter your answer in the box.

Part B

The hospital is 3. 1 miles west of the fire station. What is the length of a straight line between the school and the hospital? Round to the nearest

tenth. Enter your answer in the box.

Answers

Using Pythagorean theorem the distance between the school and the fire station is approximately 4.7 miles, while the distance between the school and the hospital is approximately 7.8 miles.

To solve this problem, we can use the Pythagorean theorem to find the distances and then add them up to get the total distance between the school and the fire station, and then between the school and the hospital.

Part A:

Let's call the middle school point A, Town Hall point B, and the fire station point C. We can draw a right triangle with AB as the base, BC as the height, and AC as the hypotenuse. Using the Pythagorean theorem, we have:

AC^2 = AB^2 + BC^2

AC^2 = 4.3^2 + 1.7^2

AC^2 = 18.98 + 2.89

AC^2 = 21.87

AC ≈ 4.7 miles (rounded to the nearest tenth)

Therefore, the length of the straight line between the school and the fire station is approximately 4.7 miles.

Part B:

Let's call the hospital point D. We can draw another right triangle with CD as the base, BC as the height, and BD as the hypotenuse. Using the Pythagorean theorem, we have:

BD^2 = BC^2 + CD^2

BD^2 = 1.7^2 + 3.1^2

BD^2 = 2.89 + 9.61

BD^2 = 12.5

BD ≈ 3.5 miles (rounded to the nearest tenth)

Now, we can add the distance between A and B (4.3 miles) to the distance between B and D (3.5 miles) to get the total distance between A and D:

AD ≈ 7.8 miles (rounded to the nearest tenth)

Therefore, the length of the straight line between the school and the hospital is approximately 7.8 miles.

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Express the negation of each of these statements in terms of quantifiers without using the negation symbol.
a) ∀x(x > 1)
b) ∀x(x ≤ 2)
c) ∃x(x ≥ 4)
d) ∃x(x < 0)
e) ∀x((x < −1) ∨ (x > 2))
f ) ∃x((x < 4) ∨ (x > 7))

Answers

The negation of each of these statements in terms of quantifiers without using the negation symbo

a) There exists at least one x such that x is not greater than 1.
b) There exists at least one x such that x is not less than or equal to 2.
c) For all x, x is less than 4.
d) For all x, x is greater than or equal to 0.
e) There exists at least one x such that either x is not less than or equal to -1 or x is not greater than 2.
f) For all x, x is not less than 4 and x is not greater than 7.

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A favorite activity at LNHS is throwing paper

balls into the trashcan while the teacher isn't

looking. Suppose a paper ball is shot from 5 feet

off the ground, and the paper ball reaches a

height of 10 feet after 3 seconds.

*Write the equation that models the height (h)

of the paper ball at any given second (t).

Help me!!

Answers

The equation that models the height (h) of the paper ball at any given second (t) is: [tex]h = -16t^2 + 49.67t + 5.[/tex]

To write the equation that models the height (h) of the paper ball at any given second (t), we can use the formula:

[tex]h = -16t^2 + vt + s[/tex]

where v is the initial velocity (in feet per second), s is the initial height (in feet), and t is the time (in seconds).

In this case, we know that the paper ball was shot from 5 feet off the ground, so s = 5. We also know that the paper ball reached a height of 10 feet after 3 seconds, so we can use this information to find the initial velocity:

[tex]h = -16t^2 + vt + s[/tex]
[tex]10 = -16(3)^2 + v(3) + 5[/tex]
10 = -144 + 3v + 5
149 = 3v
v = 49.67 (rounded to two decimal places)

Now we can substitute the values for v and s into the equation:

[tex]h = -16t^2 + vt + s\\h = -16t^2 + 49.67t + 5[/tex]

Therefore, the equation that models the height (h) of the paper ball at any given second (t) is:

[tex]h = -16t^2 + 49.67t + 5.[/tex]

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Quadratic function f has vertex (4, 15) and passes through the point (1, 20). Which equation represents f ?
A) f(x)= -5/9(x-4)^2+15
B) f(x)= 5/9(x-4)^2+15
C) f(x)= -35/9(x-4)^2-15
D) f(x)= 35/9(x-4)^2-15

Answers

Since quadratic function f has vertex (4, 15) and passes through the point (1, 20), an equation that represents f is: B. f(x) = 5/9(x - 4)² + 15.

How to determine the factored form of a quadratic equation?

In Mathematics, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

h and k represents the vertex of the graph.a represents the leading coefficient.

Based on the information provided above, we can determine the value of a as follows:

f(x) = a(x - h)² + k

20 = a(1 - 4)² + 15

20 = 9a + 15

a = 5/9

Therefore, the required quadratic function is given by:

f(x) = a(x - h)² + k

f(x) = y = 5/9(x - 4)² + 15

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write the equation that gives the number of tickets sold, y, as a linear function of the number of hours, i, since the tickets have been on sale. enter your
answer in the box.

Answers

The linear function representing relation of the number of tickets sold and number of hours since tickets have been on sale is given by y = 32x.

Let us consider the two points of the linear function are,

(x₁, y₁) = (2, 64) and (x₂, y₂) = (6, 192).

Use the two-point form of the equation of a line .

Equation of linear function shows relationship between number of tickets sold y and the number of hours since tickets have been on sale x

Slope of the line is,

slope = (y₂ - y₁) / (x₂ - x₁)

= (192 - 64) / (6 - 2)

= 128 / 4

= 32

Apply the point-slope form of the equation of a line with the point (2, 64),

y - y₁= m(x - x₁)

⇒ y - 64 = 32(x - 2)

⇒ y - 64 = 32x - 64

⇒ y = 32x

Check if the equation of the line passes through the third point (18, 576),

y = 32x

⇒576 = 32(18)

⇒576 = 576

Since the equation of the line passes through all three points,

Therefore, linear function between the number of tickets sold and number of hours since tickets have been on sale is y = 32x.

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The above question is incomplete, the complete question is:

A linear function has the table of values shown. The information in the table shows the number of tickets sold on opening night of a movie as a function of the number of hours since the tickets have been on sale.

Number of Hours (x) 2 6 18

Number of Tickets Sold (y) 64 192 576

Write the equation that gives the number of tickets sold, y, as a linear function of the number of hours, x, since the tickets have been on sale. Enter your answer in the box

In the diagram of circle A shown below , chords CD snd EF intersect at G, and chords CE and FD and drawn

Which statements is not always true?

Answers

The incorrect statement about the intersecting triangles is A. CG ≅ FG.

Why is the statement CG ≅ FG incorrect about the intersecting triangles?

With intersecting triangles, it is not always guaranteed that segments like CG and FG will be congruent. The lengths of CG and FG will depend on the specific configuration of the chords and their intersection point G.

However, CE/EG = FD/DG statement is TRUE. This is a consequence of the Intersecting Chords Theorem. When two chords intersect inside a circle, the products of their segments are equal.

Since ∠CEG ≅ ∠FDG intersect inside a circle, the corresponding intercepted arcs create equal angles at the intersection point. Therefore the statement is true.

ΔCEG ~ ΔFDG is also true because we know that the triangles share an angle and have proportional sides.

The answer above is in response to the full question below;

In the diagram of circle A shown below , chords CD and EF intersect at G, and chords CE and FD and drawn

Which statements is not always true?

a. CG ≅ FG

b. CE/EG = FD/ DG

c. ∠CEG ≅ FDG

d. ΔCEG ~ ΔFDG

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Your friend makes a stem-and-leaf plot of the data. 51, 25, 47, 42, 55, 26, 50, 44, 55 Student work is shown. A stem and leaf plot. A vertical line separates each stem from its first leaf. The first row has a stem of 2 and leaves 5 and 6. The second row has a stem of 4 and leaves 2, 4, and 7. The third row has a stem of 5 and leaves 0, 1, 5, and 5. The key shows 4 vertical bar 2 is equal to 42. Is your friend correct? Responses yes yes no no Question 2 Explain your reasoning.

Answers

Yes, your friend is not correct about the stem and leaf plot.

How to design the stem and leaf plot ?

The stem and leaf plot made by your friend is:

Stem | Leaves

2 | 5, 6

4 | 2, 4, 7

5 | 0, 1, 5, 5

Key : 4 | 2 = 42

When the data points from these are taken, we have :

25 , 26 , 42 , 44 , 47 , 50, 51, 55, 55

This is the same as the data provided of :

51, 25, 47, 42, 55, 26, 50, 44, 55

So, your friend's stem and leaf plot is indeed correct.

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A cube has a volume of 200cm3. What is the length of one edge?
Give your answer in cm correct to one decimal place

Answers

Answer:

The answer is 5.8cm to 1 d.p

Step-by-step explanation:

volume of cube=L³

200=L³

cube root both sides

³√200=³√L³

L=5.8cm to 1 d.p

A $70,000 mortgage is $629. 81 per month. What was the percent and for how many years?

9%, 20 years

9%, 25 years

7%, 20 years

9%, 30 years

Answers

The correct answer is 9% interest rate and 25 years.

To find the correct answer, we can use the mortgage payment formula:

M = P * (r(1 + r)^n) / ((1 + r)^n - 1)

Where:
M = monthly mortgage payment ($629.81)
P = principal loan amount ($70,000)
r = monthly interest rate (annual interest rate / 12)
n = total number of payments (years * 12)

We can test each option to see which one fits the given mortgage payment.

1) 9%, 20 years:
r = 0.09 / 12 = 0.0075
n = 20 * 12 = 240

M = 70000 * (0.0075(1 + 0.0075)^240) / ((1 + 0.0075)^240 - 1)
M ≈ $629.29 (close but not exact)

2) 9%, 25 years:
n = 25 * 12 = 300

M = 70000 * (0.0075(1 + 0.0075)^300) / ((1 + 0.0075)^300 - 1)
M ≈ $629.81 (matches the given mortgage payment)

Based on our calculations, the correct answer is 9% interest rate and 25 years.

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