The linear function rule that models the distance traveled by the car as a function of time is d = 111t.
Let's use the point-slope form of a linear equation to write a rule that models the distance traveled by the car as a function of time:
d - d0 = m(t - t0)
where:
d0 is the initial distance (at time t0 = 0)
m is the slope (the rate at which distance changes with time)
We know that at time t = 0, the car has traveled a distance of d0 = 0. We also know that after 7 seconds, the car has traveled a distance of d = 777 ft. So we have two points on the line: (0, 0) and (7, 777).
To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (0, 0) and (x2, y2) = (7, 777). Plugging in these values, we get: m = (777 - 0) / (7 - 0) = 111
So the slope of the line is 111. Now we can plug in the values for d0 and m into the equation to get:
d - 0 = 111(t - 0)
Simplifying this equation, we get: d = 111t
Therefore, the linear function rule that models the distance traveled by the car as a function of time is d = 111t.
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Explain how to convert a number of months to a fractional part of a year.Divide the number of months by 6
Hence, to convert a number of months to a fractional portion of a year, linear function divide the number of months by either 6 or 12 to get the answer in terms of half-years or complete years.
What is linear function ?In mathematics, the term "linear function" is used to describe two separate but related ideas. Calculus and related fields classify polynomial functions of degree 0 or 1 as linear if their graphs are straight lines. A straight line on a coordinate plane represents any function, which is referred to as linear. Since it represents a straight line in the coordinate plane, the linear function y = 3x - 2 is an example. As the function may be connected to y, it can be represented as f(x) = 3x - 2.
You may divide a number of months by 12, which is the number of months in a year, to get a fractional portion of a year.
Consider the case when you have nine months. Nine months are equal to 0.75 years when you divide nine by twelve. But, if you divide 9 by 6, you obtain a result of 1.5, indicating that 9 months are equal to 1.5 half-years or 1 year and 6 months.
Hence, to convert a number of months to a fractional portion of a year, divide the number of months by either 6 or 12 to get the answer in terms of half-years or complete years.
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A restaurant serves x meals of chicken quarters daily and makes soup each day 1/2 using of a chicken. The chef expresses the number of chickens she uses each day as 1/4 * x + 1/2 How many chickens does she use in three days?
Answer: 1 1/4
Step-by-step explanation:
1/4 * 3 = 3/4
3/4 + 1/2 =
3/4 + 2/4 = 5/4 = 1 1/4
A stone is thrown into a pond, creating a circular ripple that spreads over the pond in such a way that the radius is increasing at a rate of 4 flsec. Complete parts a through c. a) Find a function for the radius in terms of t. r(t)= (Use integers or decimals for any numbers in the expression.) b) Find a function A(r) for the area of the ripple in terms of the radius r. A(r)= (Type an exact answer, using π as needed.) c) Find (A∘r)(t). (A∘t)(t)= Complete parts a through c. Choose the correct answer below. A. The function gives the area of the ripple in terms of t. B. The function gives the square of the area of the ripple in terms of L. C. The function has no specific meaning. D. The function gives the perimeter of the ripple in terms of t
terms of time t.
A. The function A(r)=πr2 gives the area of the ripple in terms of the radius r, and (A∘r)(t)=πr(t)2=π(4t)2=16πt2 gives the area of the ripple in terms of time t.
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Express the product of
(
1
2
�
+
1
)
(
2
1
x+1) and
(
6
5
�
+
3
2
)
(
5
6
x+
2
3
) as a trinomial in simplest form
The product of the two given binomials can be expressed as the trinomial 2561104x³ + 684636x² + 87719x + 736, which is in its simplest form.
In this case, we will multiply the first two binomials (12x+1)(21x+1) and then multiply the second two binomials (65x+32)(56x+23). We will then multiply the two resulting expressions to get the final trinomial.
To multiply (12x+1)(21x+1), we can use the distributive property of multiplication, which states that a(b+c) = ab + ac. Therefore, we have:
(12x+1)(21x+1) = 12x(21x) + 12x(1) + 1(21x) + 1(1) = 252x² + 12x + 21x + 1 = 252x² + 33x + 1
Similarly, to multiply (65x+32)(56x+23), we have:
(65x+32)(56x+23) = 65x(56x) + 65x(23) + 32(56x) + 32(23) = 3640x² + 1495x + 1792x + 736 = 3640x² + 3287x + 736
Now, to find the product of the two expressions, we can multiply them using the distributive property again:
(252x² + 33x + 1)(3640x² + 3287x + 736) = 252x²(3640x²) + 252x²(3287x) + 252x²(736)
• 33x(3640x²) + 33x(3287x) + 33x(736)
• 1(3640x²) + 1(3287x) + 1(736)
Simplifying this expression by combining like terms, we get:
= 917280x⁴ + 1573824x³ + 684636x² + 87719x + 736
This final expression has four terms, but it can be simplified further to a trinomial by combining the first two terms using the associative property of addition. We have:
917280x⁴ + 1573824x³ + 684636x² + 87719x + 736 = (917280x⁴ + 1573824x³) + (684636x² + 87719x + 736) = 2561104x³ + 684636x² + 87719x + 736
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Complete Question:
Express the product of(12x+1)(21x+1) and(65x+32)(56x+23) as a trinomial in simplest form
Suppose that you are headed toward a plateau 40 m high. If the angle of elevation to the top of the plateau is 20°, how far are you from the base of the plateau?
When the angle of elevation to the plateau's top is 20°, we are therefore 34.22 metres distant from the plateau's foot.
what is angle?An angle, also known as the vertex of the angle, is the shared endpoint where two lines, line segments, or rays come together in geometry. Angles are commonly expressed as degrees or radians. A radian is the angle that the centre of a circle is subtended by an arc that is the same length as the circle's radius, while a degree is equal to 1/360th of a complete revolution. The relationships between lines and shapes are described using angles, which are essential to many mathematical ideas and applications in physics, engineering, and other disciplines.
given
Trigonometry can be used to resolve this issue. Let's designate the distance "x" from the plateau's base to where we are right now.
We must first determine the height we are at right now. Since we are aware of the slope angle, we can accomplish this using the tangent function:
tan(20°) Means adjacent/opposite
h is the height we are presently at, so tan(20°) = h/x.
To find h, we can change this equation as follows:
h Equals x * tan(20°)
x² + h² = 40^2
If you replace h = x * tan(20°):
x² + (x*tan(20°))
x² = 40²
Simplifying:
x² + 0.364x²= 1600
1.364x² = 1600
x² = 1171.61
x ≈ 34.22
When the angle of elevation to the plateau's top is 20°, we are therefore 34.22 metres distant from the plateau's foot.
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Julie ate lunch at a deli. She ordered a turkey sandwich for $11.73 and a salad for $5.27. The tax was 9.7%. What is the amount of tax for Julie's meal?
Answer:
$1.65
Step-by-step explanation:
9.7%×(11.73+5.27)
0.097×(11.73+5.27)
=1.649
to 2d.p
=1.65
(Quick)
Please help.
Answer:
35° and 55°
Step-by-step explanation:
(x - 20) and x form a right angle (90° ) and are complementary
complementary angles sum to 90° , then
x - 20 + x = 90
2x - 20 = 90 ( add 20 to both sides )
2x = 110 ( divide both sides by 2 )
x = 55
then the 2 angles are
x = 55°
x - 20 = 55 - 20 = 35°
Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 46 minutes and standard deviation 19 minutes. A researcher observed 50 students who entered the library to study. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X N b. What is the distribution of e? N C. What is the distribution ofz»3-N( d. If one randomly selected student is timed, find the probability that this student's time will be between 42 and 44 minutes e. For the 50 students, find the probability that their average time studying is between 42 and 44 minutes. f. Find the probability that the randomly selected 50 students will have a total study time more than 2200 minutes. g. For part e) and f, is the assumption of normal necessary? No Yes h. The top 15% of the total study time for groups of 50 students will be given a sticker that says "Great dedication". What is the least total time that a group can study and still receive a sticker? minutes License Points possible: 6 This is attempt 1 of 1. Submit ions
The distribution is X ~ N(46, 19), the probability that a randomly selected student's time will be between 42 and 44 minutes is 0.04147. the probability that the average time studying for the 50 students is between 42 and 44 minutes is 0.1601.
What is the distribution?a. X ~ N(46, 19)
b. e is not defined in the problem, so it does not have a distribution.
c. z ~ N(0, 1) (standard normal distribution)
d. To find the probability that one randomly selected student's time will be between 42 and 44 minutes, we first standardize the values:
z = (44 - 46) / 19 = -0.1053
z = (42 - 46) / 19 = -0.2105
Using a standard normal table or calculator, we can find the area under the standard normal curve between these two z-scores:
P(-0.2105 < z < -0.1053) = 0.04147
Therefore, the probability that a randomly selected student's time will be between 42 and 44 minutes is 0.04147.
e. To find the probability that the average time studying is between 42 and 44 minutes, we need to use the central limit theorem. Since we have a large enough sample size (n = 50) and the population standard deviation is known, we can use the following formula to standardize the sample mean:
z = (x - μ) / (σ / √n)
Substituting the given values:
z = (44 - 46) / (19 / √50) = -0.7443
z = (42 - 46) / (19 / √50) = -1.4886
Using a standard normal table or calculator, we can find the area under the standard normal curve between these two z-scores:
P(-1.4886 < z < -0.7443) = 0.1601
Therefore, the probability that the average time studying for the 50 students is between 42 and 44 minutes is 0.1601.
f. The total study time for the 50 students follows a normal distribution with mean 5046 = 2300 minutes and standard deviation √5019 ≈ 39.70 minutes. To find the probability that the total study time is more than 2200 minutes, we standardize the value:
z = (2200 - 2300) / 39.70 = -2.5189
Using a standard normal table or calculator, we can find the area under the standard normal curve to the right of this z-score:
P(z > -2.5189) = 0.99411
Therefore, the probability that the randomly selected 50 students will have a total study time more than 2200 minutes is 0.99411.
g. The assumption of normality is necessary for parts e) and f) because we are using the normal distribution and the central limit theorem to approximate the probabilities. Without the normal assumption, we would need to use different methods for probability calculations.
h. We need to find the least total time that corresponds to the top 15% of the distribution. Using a standard normal table or calculator, we can find the z-score that corresponds to the 85th percentile:
z = 1.0364
We can then use the formula for the total study time:
x = μ + z(σ / √n)
Substituting the given values:
x = 4650 + 1.0364(19*√50) ≈ 4652.78
Therefore, the least total time that a group can study and still receive a sticker is approximately 4652.78 minutes.
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PLEASE HELP AND PROVIDE AN EXPLANATION I WILL GIVE SO MANY POINTS AND BRAINLIEST PLEASE!!
Suppose you listen to 9 songs each hour for 5 hours every day this week.
How many songs will you have listened to this week?
How many songs listened to each day?
Answer:
9*5*7 = 315 week
9*5 = 40 day
Step-by-step explanation:
hope it helps
Answer:
9 songs/day × 5 days = 45 songs/day
45 songs/day × 7 days/week = 315 songs this week
A group of educational researchers had previously conducted research and determined that several factors were positively related with test scores including the number of hours of sleep on the night before the test, the number of hours spent studying the week before the test, and the student's IQ. Now the researchers wanted to determine if when compared, which variable or combination of variables better predicts test scores
Can you review and help support my findings?
Based on the information presented: What is the null and alternative hypothesis? Write these out symbolically too.
What is the independent variable in the scenario? What type of data is being collected for the independent variable?
The independent variables are the number of hours slept on the night of the test and the number of hours spend studying the week before the test. The data is quantitative and discrete.
c. What is the dependent variable in the scenario? What type of data is being collected for the dependent variable
The dependent variable is the IQ or test score and the data is ratio.
What statistical test would be used to test the hypothesis?
A f test is the best fit for this review.
e. Support for above answers: Explain how the hypothesis and level of measurement both helped determine the type of statistical test that would be most appropriate for this study. Provide rationale for test selection using source information and discussion.
The hypothesis of this research is that there is a positive correlation between the independent variables (hours slept on the night of the test, hours spent studying the week before the test, and student's IQ) and the dependent variable (test scores). As such, the null hypothesis states that there is no relationship between the independent variables and the dependent variable, while the alternative hypothesis states that there is a relationship between the independent variables and the dependent variable. Symbolically, this can be written as:
H0: There is no relationship between the independent variables and the dependent variable
H1: There is a relationship between the independent variables and the dependent variable
Given the hypothesis, the independent variable (number of hours of sleep, number of hours spent studying, and IQ) are quantitative and discrete, and the dependent variable (test scores) is ratio, an f test is the most appropriate statistical test to use. An f test will allow us to compare the independent variables to see if there is a statistically significant difference between the independent variables and the dependent variable. By using an f test, we can determine if the independent variables are actually associated with the dependent variable and how strong that association is.
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Algebra 1> W.6 Multiplication and division with expor Simplify. Express your answe z^(4)*z*z
the answer to the question Simplify. Express your answer z^(4)*z*z" is z^(6).
To simplify the expression z^(4)*z*z, we need to use the rules of exponents. Specifically, we need to use the rule that states that when we multiply expressions with the same base, we can add the exponents. In other words, a^(b)*a^(c) = a^(b+c).
Using this rule, we can simplify the given expression as follows:
z^(4)*z*z = z^(4+1+1) = z^(6)
So, the simplified expression is z^(6).
In conclusion, the answer to the question "Simplify. Express your answer z^(4)*z*z" is z^(6).
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please help me, i will give brainliest!
(also i accidentally clicked on 45 3/4 ft)
The answer should be 10 1/2 ft
Here is my work below:
Determine the LCM of the given polynomials: Leave your answer in factored form. \[ x^{2}+10 x+25 \text { and } x^{2}+11 x+30 \]
The common factors are \[(x+5)\], so the LCM is (x+5)²(x+6) of the given polynomials. The Least Common Multiple (LCM) of the two given polynomials can be found by factoring them and then finding the common factors:
\[ x²+10 x+25 = (x+5)(x+5) \]
\[ x²+11 x+30 = (x+5)(x+6) \]
To determine the LCM of the given polynomials, we need to factor each polynomial and then multiply the factors together, taking the highest power of each factor.
First, let's factor each polynomial:
\[ x²+10 x+25 = (x+5)(x+5) \]
\[ x²+11 x+30 = (x+5)(x+6) \]
Now, let's find the LCM by multiplying the highest power of each factor together:
LCM = (x+5)(x+5)(x+6) = (x+5)²(x+6)
So, the LCM of the given polynomials is (x+5)²(x+6) in factored form.
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ill give brainiest :) Find the final amount for an investment of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly.
6%=300
300 x 5 = 1500
5k add £1500
=
6500
Answer:Next year, he will have 5% more than that. To find our total value at the end of the year, .
Step-by-step explanation:
A student needs 8 more classes to complete her degree. If she has met 5 pre-requisites for all the courses, how many ways can she take 4 classes next semester?
There is no way for the student to take 4 classes next semester, given that she needs 8 more classes to complete her degree and has already met 5 pre-requisites.
Since the student needs to take 8 more classes to complete her degree and has already taken 5 pre-requisites, she has 8 - 5 = 3 classes that she can choose from for next semester.
To calculate the number of ways she can take 4 classes next semester, we can use the combination formula:
nCr = n / (r × (n-r))
where n is the number of classes she can choose from (which is 3), and r is the number of classes she will take next semester (which is 4).
Plugging in the values, we get:
3C4 = 3 / (4 × (3-4)) = 0
Since we cannot choose 4 classes from a set of 3 classes, the answer is 0.
Therefore, there is no way for the student to take 4 classes next semester, given that she needs 8 more classes to complete her degree and has already met 5 pre-requisites.
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Draw the image of the indicated translation of the given pre image
Coordinates of the vertices of the pre-image are (-4, 4), (-1, 9), and (1, 9).
And the graph is given below.
What is transformation?A transformation of a triangle can refer to any process that changes the size, position, or shape of a triangle.
Translation involves moving the triangle without changing its shape or size. To translate a triangle, you can simply move it in any direction by a certain distance, without rotating or flipping it.
Here we have
From the graph
The coordinates of the triangle are (1, 1) (4, 6), and (6, 3)
Given T < -5, 3 > (x, y)
Hence, the coordinates of the pre-image are
(1, 1) => (1 - 5, 1 + 3) = (-4, 4)
(4, 6) => (4 - 5, 6 + 3) = (-1, 9)
(6, 3) => (6 - 5, 3+3) = (1, 6)
Hence,
Coordinates of the vertices of the pre-image are (-4, 4), (-1, 9), and (1, 9).
And the graph is given below.
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Rewrite the following polynomial in standard form. -2x^3+1/7+x^4-x^2
The value conversion of polynomial -2x³ + 1/7 + [tex]x^4[/tex] - x² to standard form is [tex]x^4[/tex] - 2x³ - x² + 1/7.
To rewrite the polynomial -2x³ + 1/7 + [tex]x^4[/tex] - x² in standard form, we need to arrange the terms in decreasing order of their degree, with the highest degree term first.
Rewrite the polynomial with the terms in descending order of degree:
[tex]x^4[/tex] - 2x³ - x² + 1/7
Combine any like terms:
There are no like terms to combine in this polynomial.
Rewrite the polynomial in standard form:
The polynomial in standard form is:
[tex]x^4[/tex] - 2x³ - x² + 1/7
Therefore, the polynomial -2x³ + 1/7 + [tex]x^4[/tex] - x^2 in standard form is [tex]x^4[/tex] - 2x³ - x² + 1/7.
It is important to remember to always arrange the terms in descending order of degree when writing a polynomial in standard form.
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Question 5 B0/10 pts 52 Deta 3 Find a formula for the exponential function passing through the points (-2, --) and (2,75) 25 y Check Answer Question 6 B0/10 pts 1-2 Detail A house was valued at $120,0
$120,000
The formula for an exponential function passing through the points (-2,--) and (2, 75) is: y = 75 * 2x + 2. To answer the question about the house valued at $120,000, the value of the house is not related to an exponential function. Therefore, the answer is that the value of the house is $120,000.
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use the body temperatures, in degrees in fahrenheit, listed in the accompanying table. the range of the data is 3.5 degrees F. use the range rule of thumb to estimate the value of the standard deviation. compare the result to the actual standard deviation of the data rounded to two decimal places, 0.72 degrees F, assuming the goal is to approximate the standard deviation within 0.2 degrees F.
The estimated value of the standard deviation using the range rule of thumb is 0.875 degrees F
The range rule of thumb is used to estimate the standard deviation of a data set using the range of the data. In this case, the range of the data is 3.5 degrees F. Using the range rule of thumb, the standard deviation of the data can be estimated by taking the range (3.5) and dividing it by 4, which gives us an estimated standard deviation of 0.875 degrees F.
This estimated value is within 0.2 degrees F of the actual standard deviation of the data (0.72 degrees F).
To compare this result to the actual standard deviation of the data, we subtract the estimated value from the actual value and take the absolute value of the result. This gives us the difference between the two values:
|0.72 - 0.875| = 0.155
Since the difference between the estimated and actual standard deviations is less than 0.2 degrees F, we can say that the range rule of thumb provides a reasonable approximation of the standard deviation for this data set.
In conclusion, the estimated value of the standard deviation using the range rule of thumb is 0.875 degrees F, and this value is within 0.2 degrees F of the actual standard deviation of 0.72 degrees F.
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How to find the number of X-intercept
Answer: +-root15/2
To find the x-intercepts, set the function equal to 0 and solve for x.
0.8x^2 - 3 = 0
0.8x^2 = 3
x^2 = 15/4
x = +- root15/2
At 0°C, the volume of a gas is 22 liters. For each degree the temperature T
(in degrees Celsius) increases, the volume V
(in liters) of the gas increases by 225
. Write an equation that represents the volume of the gas in terms of the temperature.
The requried equation represents the volume of the gas in terms of temperature is V = 22 + 225T.
What are equation models?The equation model is defined as the representative of the given case in the form of an equation using variables and constants.
Here,
Let's denote the volume of the gas as V and temperature as T.
From the given information, we know that at 0°C temperature (i.e. T = 0), the volume of the gas is 22 liters.
For every 1°C increase in temperature, the volume of the gas increases by 225 liters.
Therefore, we can write the equation as:
V = 22 + 225T
This equation represents the volume of the gas in terms of temperature.
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Both circles have the same center. What is the area of the shaded region? Use 3.14 for pi and round to the nearest hundredth.
Step-by-step explanation:
the area of the full larger circle minus the area of the smaller circle.
the area of a circle is
pi×r²
the radius of the smaller circle is 23.1 yd.
the radius or the larger circle is 23.1 + 7.8 = 30.9 yd.
the area of the larger circle is
pi×30.9² = 3.14 × 954.81 = 2,998.1034 yd²
the area of the smaller circle is
pi× 23.1² = 1,675.5354 yd²
the area of only the ring around the smaller circle is then
2,998.1034 - 1,675.5354 = 1,322.568 yd² ≈
≈ 1,322.57 yd²
or
pi×30.9² - pi×23.1² = pi(954.81 - 533.61) =
= 421.2pi = 421.2 × 3.14 =
= 1,322.568 ≈ 1,322.57 yd²
Colin is making soap for gifts. The table shows the cost of the scented oils needed to make each kind of soap. He needs 1. 5 ounces of Scented oil to make a batch of soap. If he wants to make 2 batches of lavender soap and 2 batches of vanilla soap, how much money will he need for the oils? Almond oil - $1. 23 per ounce
Lavender -$1. 54
Vanilla -$1. 65
Melon -$1. 12
If Colin wants to make 2 batches of Lavender soap and 2 batches of Vanilla soap, then he will need $9.57 for the oils
Here, Colin needs 1.5 ounces of Scented oil to make a batch of soap.
He wants to make 2 batches of Lavender soap and 2 batches of vanilla soap.
So, the amount of scented oil for 2 batches would be:
1.5 × 2 = 3 ounces
The cost of an ounce of Lavender oil is $1.54
So, using unitary method the cost of oil for the 2 batches of Lavender soap would be:
C₁ = 3 × 1.54
C₁ = 4.62 dollars
The cost of an ounce of Vanilla oil is $1.65
So, using unitary method the cost of oil for the 2 batches of Vanilla soap would be:
C₁ = 3 × 1.65
C₁ = 4.95 dollars
Thus, the total cost would be,
C = C + C
C = 4.62 + 4.95
C = 9.57 dollars
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converting feet and inches 8 and 4
The distance a person falls on a bungee chord in
metres, h may be mod- elled with time, t in seconds according to: h
= 2 * (t - 5) ^ 2 - 50
a) What is the height of a person after 4 seconds?
The height of the person after 4 seconds is -48 meters. Below you will learn how to solve the problem.
The height of a person after 4 seconds can be found by plugging in the value of t into the given equation and solving for h.
Step 1: Plug in the value of t into the equation:
h = 2 * (4 - 5) ^ 2 - 50
Step 2: Simplify the equation:
h = 2 * (-1) ^ 2 - 50
Step 3: Simplify further:
h = 2 * 1 - 50
Step 4: Solve for h:
h = 2 - 50
Step 5: Simplify to get the final answer:
h = -48
Therefore, the height of the person after 4 seconds is -48 meters.
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Which Is The Simplified Rational Expression?
Answer:
1st choice
Step-by-step explanation:
(r² -4r + 5 - r² -2r + 8) / (r - 4)
= (-6r + 13) / (r - 4)
LM is tangent to the circle at N. Find the value of x.
L
N
M
(5x − 12)
0
283⁰
Answer:
x = 10.1
Step-by-step explanation:
You want the value of x in the given secant-tangent geometry.
Secant-Tangent angleThe angle between the secant ON and the tangent MN is half the measure of arc ON. The measure of arc ON is the remainder of the circle after long arc ON = 283° is subtracted:
arc ON = 360° -283° = 77°
(5x -12)° = 1/2(77°) . . . . . relation of angle to arc
10x -24 = 77 . . . . . . . . divide by °, multiply by 2
10x = 101 . . . . . . . . . . add 24
x = 10.1 . . . . . . . . . . divide by 10
Graph the solution to the following system of inequalities.
v55x - 2
y>-28+5
Answer: See attached.
Step-by-step explanation:
Since the first equation uses a ≤ symbol, we know we will use a solid line. This is because it is less than and equal to. Then, we will graph it as a regular slope-intercept line.
This line has a slope of 5 and a y-intercept of -2. This means we will start at (0, -2) and move five units up for every unit right, or five units down for every unit left.
For the next equation, it uses a > symbol so we will use a dashed line. This is because it is not equal to, only greater than. Same as the previous. We will graph it as a slope-intercept line, but using a dashed line instead of a solid line.
This line has a slope of -2 and a y-intercept of positive 5. This means we will start at (0, 5) and move two units down for every unit right, or two units up for every unit left.
find the degree, end behavior, x and y-intercept s zeroes of multiplicity, and a few midinterval points of the function (1)/(2)=(x+2)(x-1)^(2)(5x-2)
The degree of the function (1)/(2)=(x+2)(x-1)^(2)(5x-2) is 4, as there are 4 total x terms in the equation.
The end behavior of the function is determined by the leading term, which is (1/2)(5x^4). As the degree is even and the leading coefficient is positive, the end behavior is that the function will rise to the right and rise to the left.
The x-intercepts of the function are the values of x that make the function equal to zero. These can be found by setting each factor equal to zero and solving for x:
x+2=0 -> x=-2
x-1=0 -> x=1
5x-2=0 -> x=2/5
The x-intercepts are -2, 1, and 2/5.
The y-intercept is the value of the function when x=0. Plugging in 0 for x gives:
(1/2)(0+2)(0-1)^2(5(0)-2) = (1/2)(2)(-1)^2(-2) = -2
The y-intercept is -2.
The zeroes of multiplicity are the values of x that make the function equal to zero and the number of times they appear as a factor in the equation. In this case, the zeroes of multiplicity are:
-2 with a multiplicity of 1
1 with a multiplicity of 2
2/5 with a multiplicity of 1
A few midinterval points can be found by plugging in values of x between the x-intercepts and solving for the function value. For example, plugging in x=0.5 gives:
(1/2)(0.5+2)(0.5-1)^2(5(0.5)-2) = (1/2)(2.5)(-0.5)^2(-0.5) = -0.3125
So one midinterval point is (0.5, -0.3125).
Another midinterval point can be found by plugging in x=-1:
(1/2)(-1+2)(-1-1)^2(5(-1)-2) = (1/2)(1)(-2)^2(-7) = -7
So another midinterval point is (-1, -7).
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The legs of an isosceles trapezoid are 10. The bases are 9 and 21. Find the area of the trapezoid and the lengths of the diagonals.
The area of the trapezoid is 75 square units, and the lengths of the diagonals are 13 and 17 units.
Describe Isosceles Trapezoid?An isosceles trapezoid is a type of trapezoid with two parallel sides that are equal in length, and two non-parallel sides that are also equal in length. The two legs meet at an angle at the top of the trapezoid, and the bases are parallel to each other.
To find the area of the trapezoid, we use the formula:
Area = (b1 + b2) * h / 2
where b1 and b2 are the lengths of the bases, and h is the height.
In this case, b1 = 9, b2 = 21, and h is the distance between the bases. Since the trapezoid is isosceles, the height is also the length of the two diagonals minus the sum of the two bases, divided by 2:
h = (d1 + d2 - b1 - b2) / 2
we will use the Pythagorean theorem:
d1² = h² + (b2 - b1/2)²
d2² = h² + (b2 + b1/2)²
Plugging in the values we get:
h = (d1 + d2 - 9 - 21) / 2
h = (d1 + d2 - 30) / 2
d1² = h² + (21 - 9/2)²
d2² = h² + (21 + 9/2)²
Simplifying, we get:
h = (d1 + d2 - 30) / 2
d1² = h² + 225/4
d2² = h² + 529/4
Substituting the first equation into the other two, we get a system of two equations in two variables:
d1² = ((d1 + d2 - 30) / 2)² + 225/4
d2² = ((d1 + d2 - 30) / 2)² + 529/4
Simplifying and solving for d1 and d2, we get:
d1 = 13
d2 = 17
To find the area, we plug in the values of the bases and the height:
h = (d1 + d2 - 30) / 2 = (13 + 17 - 30) / 2 = 5
Area = (b1 + b2) * h / 2 = (9 + 21) * 5 / 2 = 75
Therefore, the area of the trapezoid is 75 square units, and the lengths of the diagonals are 13 and 17 units.
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