Answer:
Hello! Yes, when completing the square, you are essentially transforming a quadratic equation from standard form to vertex form. The vertex form of a quadratic equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola and 'a' is a constant that determines the shape and direction of the parabola.
To convert a quadratic equation from standard form (ax^2 + bx + c) to vertex form, we follow these steps:
1. Divide both sides of the equation by 'a' to make the coefficient of x^2 equal to 1.
2. Move the constant term (c/a) to the right-hand side of the equation.
3. Complete the square by adding and subtracting (b/2a)^2 on the left-hand side of the equation.
4. Factor the left-hand side of the equation as a perfect square trinomial.
5. Write the equation in vertex form by identifying the values of 'a', 'h', and 'k'.
Consider parallelogram ABCD.
A parallelogram A B C D has the following angles labeled. D left parenthesis 85 plus y right parenthesis degrees, B left parenthesis 3 y minus 15 degrees right parenthesis degrees. The sides are labeled x minus 8 and 40 minus 2 x.
Which equation is made true by the opposite angles theorem?
A.
x − 8 = 40 − 2x
B.
40 − 2x = 85 + y
C.
x − 8 = 3y − 15
D.
3y − 15 = 85 + y
An equation which is made true by the opposite angles theorem include the following: D. 3y − 15 = 85 + y.
What is a parallelogram?In Mathematics, a parallelogram simply refers to a four-sided geometrical figure (shape) and it can be defined as a type of quadrilateral and two-dimensional geometrical figure that is composed of two (2) equal and parallel opposite sides.
According to the opposite angles theorem, the opposite angles of a parallelogram are always congruent or equal and as such, we have the following:
(3y − 15)° = (85 + y)°
Next, we would determine the value of y as follows;
(3y − 15)° = (85 + y)°
3y - y = 85 + 15
2y = 100
y = 100/2
y = 50.
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Answer:
Step-by-step explanation:
the answer is [3y-15=85+y]
Anna borrowed 20000 at 5% for half year. She want to pay 8000 on maturity. To achieve this she is planning to pay 2000 in 10 months, 5000 in 16 months from now. How much should she pay in 2 and a half years from now to meet her obligation
Anna should pay £3666.67 in 2 and a half years from now to meet her obligation. It means simple interest is £3666.67.
To solve this problem, we can use the formula for simple interest:
I = Prt
where I is the interest, P is the principal, r is the interest rate, and t is the time in years.
Given that Anna borrowed 20000 at 5% for half a year, we can calculate the interest as:
I = Prt = 20000 x 0.05 x 0.5 = 500
So, the total amount Anna needs to pay at maturity is 20000 + 500 = 20500.
Anna plans to pay 8000 on maturity, 2000 in 10 months, and 5000 in 16 months. Therefore, the remaining amount she needs to pay is:
20500 - 8000 - 2000 - 5000 = 5500
To find out how much she needs to pay in 2 and a half years from now, we need to first calculate the time in years:
2 and a half years = 2.5 years
The total time elapsed from the time of borrowing is:
0.5 year (at borrowing) + 2.5 years = 3 years
Using the formula for simple interest, we can calculate the principal amount that will accrue an interest of 5500 in 3 years at 5% per annum:
I = Prt
5500 = P x 0.05 x 3
P = 3666.67
Therefore, Anna should pay £3666.67 in 2 and a half years.
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(HELP!!) Express the trig ratios as fractions in simplest terms.
The answers for the following questions and fill uos are given below respectively.
Define the term trigonometry?Study of correlations between triangles' sides and angles is the focus of the mathematical field of trigonometry.
It solves mathematical problems, especially those involving geometric shapes and motion, by using the properties of triangle angles and sides.
We can use the given side lengths of the right triangle KJL to determine the trigonometric ratios:
sin K = opposite/hypotenuse = KJ/LK = 9/10
cos L = adjacent/hypotenuse = KJ/LK = 9/10
To express these trig ratios as fractions in simplest terms, we can simplify each fraction by dividing both the numerator and denominator by their greatest common factor, which is 1 in this case. Therefore:
sin K = 9/10
cos L = 9/10
Since sin K is the ratio of the opposite side (KJ) to the hypotenuse (LK), it corresponds to angle K. Similarly, cos L is the ratio of the adjacent side (KJ) to the hypotenuse (LK), it corresponds to angle L.
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The ratio of the opposing side (KJ) to the hypotenuse is called sin K, which is equivalent to angle K. (LK). Cos L is the ratio of the neighboring side (KJ) to the hypotenuse and is related to angle L. (LK).
Define the term trigonometry?The area of mathematics known as trigonometry focuses on the analysis of relationships between shapes' sides and angles.
It uses the characteristics of triangle angles and sides to answer mathematical problems, particularly those involving geometric shapes and motion.
The trigonometric ratios can be calculated using the side lengths of the right triangle KJL as given:
sin K = opposite/hypotenuse = KJ/LK = 9/10
cos L = adjacent/hypotenuse = KJ/LK = 9/10
We can simplify each fraction by dividing the numerator as well as the denominator by their greatest common factor, which is 1 in this instance, to express these trig ratios as fractions in the simplest possible terms.
sin K = 9/10
cos L = 9/10
Sin K correlates to angle K because it is the ratio of the opposite side (KJ) to the hypotenuse (LK). Similar to cos L, which correlates to angle L, cos L is the ratio of the adjacent side (KJ) to the hypotenuse (LK).
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Surface area of cylinder in terms of pi? Diameter of circle 3.4 inches. Height of rectangle 3 in
Surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.
To find the surface area of a cylinder in terms of pi, you need to add the areas of the top and bottom circles (which are identical) and the area of the curved lateral surface.
The formula for the area of a circle is [tex]A = \pi r^2[/tex], where r is the radius of the circle. Since the diameter of the circle is given as 3.4 inches, we can find the radius by dividing the diameter by 2:
r = 3.4 / 2 = 1.7 inches
Using this radius, the area of one circle is:
A = [tex]\pi (1.7)^2[/tex] = [tex]9.08\pi[/tex] square inches
Since the cylinder has two identical circles, the total area of both circles is:
2A = 2([tex]9.08\pi[/tex]) = [tex]18.16\pi[/tex] square inches
The lateral surface area of a cylinder can be found by multiplying the circumference of the base circle by the height of the cylinder. The formula for the circumference of a circle is C = 2πr, so the circumference of our circle is:
C =[tex]2\pi (1.7)[/tex]= [tex]3.4\pi[/tex] inches
The height of the cylinder is given as 3 inches, so the lateral surface area is:
A = Ch = ([tex]3.4\pi[/tex])(3) = [tex]10.2\pi[/tex] square inches
To find the total surface area of the cylinder, we add the areas of the two circles and the lateral surface:
Total surface area = 2A + A = [tex]18.16\pi + 10.2\pi[/tex] = [tex]28.36\pi[/tex] square inches
Therefore, the surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.
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Find the volume of the rectangular prism.
10 m
Not drawn to scale.
21 m³
42 m³
240 m³
480 m³
Find the cross product a ⨯
b. A = 4, 5, 0 , b = 1, 0, 3 verify that it is orthogonal to both a and
b. (a ⨯
b. · a = (a ⨯
b. · b =
We can verify that a ⨯ (a ⨯ b) is orthogonal to a. We can solve it in the following way.
To find the cross product a ⨯ b, we can use the formula:
a ⨯ b = |a| |b| sinθ n
where |a| and |b| are the magnitudes of vectors a and b, θ is the angle between a and b, and n is a unit vector perpendicular to both a and b, determined by the right-hand rule.
Using the given vectors a and b, we have:
|a| = √(4² + 5² + 0²) = √41
|b| = √(1² + 0² + 3²) = √10
To find the angle between a and b, we can use the dot product:
a · b = |a| |b| cosθ
4(1) + 5(0) + 0(3) = √41 √10 cosθ
Simplifying:
4 = √410 cosθ
cosθ = 4/√410
θ ≈ 77.09°
Now we can find the cross product using the formula:
a ⨯ b = |a| |b| sinθ n
a ⨯ b = √41 √10 sin(77.09°) n
Using the right-hand rule, we can determine that the unit vector n points in the negative z-direction, so n = -k, where k is the unit vector in the positive z-direction.
Thus, we have:
a ⨯ b = -√410 sin(77.09°) k
Evaluating this expression, we get:
a ⨯ b ≈ -0.667i + 1.777j - 3.617k
To verify that this vector is orthogonal to both a and b, we can take the dot product of a ⨯ b with a and b separately. If the dot product is zero, then the vectors are orthogonal.
a ⨯ b · a = (-0.667)(4) + (1.777)(5) + (-3.617)(0) = 0
a ⨯ b · b = (-0.667)(1) + (1.777)(0) + (-3.617)(3) = 0
Since both dot products are zero, we can conclude that the vector a ⨯ b is orthogonal to both a and b.
Note: The dot product of a vector and its own cross product is always zero, which can be used to verify that a ⨯ (a ⨯ b) is orthogonal to a. In this case, we have:
a ⨯ (a ⨯ b) = (4, 5, 0) ⨯ (-0.667, 1.777, -3.617)
= -17.07i - 6.248j - 12.005k
(a ⨯ (a ⨯ b)) · a = (-17.07)(4) + (-6.248)(5) + (-12.005)(0) = 0
Thus, we have verified that a ⨯ (a ⨯ b) is orthogonal to a.
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Find the area of the shaded segment of the circle
I need more info for your problem :
Maybe this could help you :Assume a sector having a radius
Assume a sector having a radius r and a central angle of a degrees.The sector's area will be [tex]A^{2}[/tex] = (a / 360 °)* [tex]\pi[/tex]*[tex]r^{2}[/tex]
The area of an equilateral triangle having a side lenght a is A =( [tex]\sqrt{3}[/tex]/4) * [tex]a^{2}[/tex]
Writing and evaluating a function that models a real world
Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.
what is unitary method ?To answer mathematical issues involving proportional relationships between two or more quantities, one uses the unitary method. It is a technique for determining the value of one unit depending on the value of another, in other words. In several disciplines, including mathematics, physics, economics, and engineering, the unitary technique is frequently utilized. Finding the value of a single unit of a given quantity and using that value to calculate the value of a desired number of units is what this process entails.
given
The initial amount of water plus the water that has been added over time equals the total amount of water in the pond at any one time. We can apply the following formula because the rate of water addition is constant at 28 liters per minute:
W = 600 + 28T
where W is the pond's total volume of water (measured in liters), and T is the passing of time (in minutes).
T = 12 can be used in the equation to determine how much water was in the pond overall after 12 minutes:
W = 600 + 28T
W = 600 + 28(12) (12)
W = 600 + 336
W = 936
Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.
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The complete question is:- Writing and evaluating a function that models a real-world...
Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 28 liters per minute. There are 600 liters in the pond to start. Let W represent the total amount of water in the pond (in liters), and let 7' represent the total number of minutes that water has been added. Write an equation relating W to T. Then use this equation to find the total amount of water after 12 minutes.
. Create a box plot using 28, 34, 40, 24, 44, 49, and 32 as the data
ps: can you please draw a diagram and upload the pic here? i need it ASAP
Anton borrowed some money from his parents. Every week he earned $23 from doing chores. After 12 weeks he had $200 after paying back his parents.
a) how much did anton borrow from his parents
b) write an equation to represent the amount of money,y,anton will have after x weeks
Answer:
a. $76.00
b. y = 23x - 76
Step-by-step explanation:
a. If Anton earned $23 every week for twelve weeks, his total revenue would be 23 (12) = 276 dollars. He only had $200 after paying his parents back, though, so that means if you take away the $200 leftover money he got (after paying his parents back) from the $276 he earned over the 12 weeks, you'll get 276 - 200 = he borrowed $76.00 from his parents.
b. Anton is earning $23 every week, so the money, y, that he'll have after x weeks is y = 23x. But since he borrowed $76 from his parents, that much money should be subtracted from his money-making equation, leaving us with y = 23x - 76. Hope this was helpful!
what ratio is commonly used as an alternative to 3.14?
Answer:
it is typically 22 to 7
Step-by-step explanation:
if you divide 22 by 7 you get 3.142857143...
Answer:
22/7
Step-by-step explanation:
You want to know a ratio commonly used as an alternative to 3.14 for an approximation to pi.
ApproximationsThe value 3.14 as an approximation of pi is obtained by simply truncating the value to 2 decimal places. That makes the approximation lower than the actual value by about 0.0507%.
Rational approximations can be found from the expression of pi as a continued fraction. Truncating that fraction at various points gives the approximations ...
22/7 . . . . high by about 0.0402%
333/106 . . . . low by about 0.0026%
355/113 . . . . . high by about 0.0000085%
22/7 is the most commonly used rational approximation of pi instead of 3.14.
__
Additional comment
The continued fraction is non-repeating. It starts ...
[tex]3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\dots}}}}}[/tex]
2. Minor arc PQ in circle O below is 27. If angle PQO is 45°, what is the area of the shaded region?
Α) 2π
B) 4
C) 8
D) 16
The radius of the circle is approximately 16 units.
Define the term Circle identities?The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.
Here, ∠PQO = 45° So, ∠QPO = 45° (because OP = OQ radius)
So, central angle is 90°
The formula of arc length of a circle is :
arc length = [tex]\frac{central angle}{360} * (2\pi r)[/tex] ; here, the radius of the circle is r
Substituting the values we get:
27 = (90/360) x (2πr)
r = 54/π
Therefore, the radius of the circle is approximately 17.1 units.
by using radius we can find any area in circle.
So, nearby option D. is correct 16 units.
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Identify the key features of the exponential function f(x) = 5x and its graph by completing each sentence. please, i have a test tomorrow
The graph of the exponential function f(x) = [tex]5^{x}[/tex] is an upward-sloping curve that starts at (0, 1) and gets steeper and steeper as x increases. It never touches or crosses the x-axis, but it gets arbitrarily close to it as x approaches negative infinity.
EquationsAs we may raise 5 to any power, all real numbers (-∞,∞ ) are the function's domain (positive, negative, or zero).
Range: The range of the function is all positive real numbers (0, ∞), since the exponential function always gives a positive result.
Asymptote: The graph of the function approaches the x-axis as x approaches negative infinity. Therefore, the x-axis is the horizontal asymptote.
Y-intercept: The y-intercept of the function is (0, 1), since[tex]5^{0}[/tex] = 1.
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A parallelogram is shown below. Work out the size of the angle marked b. 44° b
Answer:
b = 136°
Step-by-step explanation:
b + 44 = 180°
180° - 44° = 136°
Question Progress
Homework Progress
12/
Expand and simplify 2(x + 7) + 3(x + 1)
Answer:
Step-by-step explanation:
2x + 14 + 3x +3
17 + 5x
x=3.4
a sample of 5 months of sales data provided the following information. month 1 2 3 4 5 units sold 99 100 87 92 92 (a) develop a point estimate of the population mean number of units sold per month. (b) develop a point estimate of the population standard deviation. (round your answer to two decimal places.)
Answer:
The answer would be
Step-by-step explanation:
A. develop a point estimate of the population mean number of units sold per month.
I must say, I do hope this helps so good luck dude
Amanda trains to run a marathon. She must follow a strict schedule to make sure she is ready for the race. She will start her training by running two miles the first week. She wants to run one fewer mile the next week, and then three more miles the week after that. She will continue this pattern during her entire training regimen. Write a sequence for the number of miles that Amanda will run the first 10 weeks of her training. Explain your reasoning.
The sequence for the number of miles that Amanda will run in the first 10 weeks of her training is
2, 1, 5, 0, 8, 3, 11, 5, 14, 7
What is a sequence?
A sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms).
To find the sequence for the number of miles that Amanda will run in the first 10 weeks of her training, we need to first identify the pattern of increasing and decreasing miles she will run each week.
According to the given information, Amanda will start by running 2 miles in the first week.
In the second week, she will run one fewer mile, so she will run 2 - 1 = 1 mile. In the third week, she will run three more miles than she did in the first week, so she will run 2+3=5 miles.
In the fourth week, she will run two fewer miles than she did in the second week, so she will run 1 - 2 = -1 mile.
However, it doesn't make sense for Amanda to run a negative number of miles, so we can assume that she will take a rest week during the fourth week and not run any miles.
Then, she will resume her pattern of increasing and decreasing miles in the following weeks.
Therefore, we can write the sequence for the number of miles that Amanda will run in the first 10 weeks of her training as follows:
2, 1, 5, 0, 8, 3, 11, 5, 14, 7
To obtain this sequence, we applied the given pattern of running one fewer mile in the second week, followed by three more miles in the third week, and then repeating this pattern. We also accounted for the rest week during the fourth week by inserting a 0 in the sequence.
Hence, the sequence for the number of miles that Amanda will run in the first 10 weeks of her training is
2, 1, 5, 0, 8, 3, 11, 5, 14, 7
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Find the surface area of a storage container with dimensions 24 feet long, 18 feet wide, and 8 feet high.
The surface area of the storage container is 1536 square feet. To find the surface area of a storage container with dimensions 24 feet long, 18 feet wide, and 8 feet high, we need to calculate the area of each of its six faces and then add them together.
The front and back face each have an area of 24 feet x 8 feet = 192 square feet.
The top and bottom faces each have an area of 18 feet x 24 feet = 432 square feet.
The left and right faces each have an area of 8 feet x 18 feet = 144 square feet.
Adding these areas together, we get:
Surface area of the container = 2(192 sq. ft.) + 2(432 sq. ft.) + 2(144 sq. ft.)
The surface area of the container = 384 sq. ft. + 864 sq. ft. + 288 sq. ft.
The surface area of the container = 1536 square feet
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What is the sum of the series?
∑j=152j
Enter your answer in the box.
Answer:
Hope this helps
Step-by-step explanation:
Sum of series is donated by the for formula below
[tex] \sf \: Sn = \frac{n}{2} [2a + (n - 1)d][/tex]
Example 3,6,9,12,...
Let's say we're finding the sum of the 20 series
n = 20, a = 3, d = 6-3 = 3
Where,
n means number of terms
a means the first term
d means the constant value between terms
S20 = 20/2[2(3) + (20-1)d]
S20 = 10[6+19(3)]
S20 = 60 + 570
S20 = 630
Voila!!
2 A pizza company bakes about 5565 pizzas over a 21-day period. If an equal number of pizzas
are baked each day, how many pizzas does the pizza company bake daily?
A 265 pizzas
B. 398 pizzas
c. 1855 pizzas
D. 2783 pizzas
Answer: "A 265 pizzas"
Step-by-step explanation:
STEP 1: Divide the number of pizzas by the number of days (5565/21).
This should give you the answer 265.
What do I do I’m confused
what is the expected value and standard deviation of the number of small aircraft that arrive during a 75-min period?
For the number of small aircraft that arrive during a 75-min period, the expected value if 10 and standard deviation is 3.17.
The expected value of the number of small aircraft that arrive during a 75-minute period is ⇒ E[X] = μ = 8t.
We need to convert the time period to hours,
So, t = 75/60 = 1.25 hours.
So, the expected number of small aircraft arrivals during a 75-minute period is ⇒ E[X] = μ = 8(1.25) = 10,
To find the standard deviation, we know that for a Poisson distribution, the standard deviation(σₓ) is the square root of mean.
So, σₓ = √μ = √8t,
Substituting t = 1.25,
We get,
⇒ σₓ = √8(1.25) = √10 ≈ 3.17,
Therefore, the expected value for number of aircraft is 10, and standard deviation is 3.17.
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The given question is incomplete, the complete question is
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α =8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t.
What are the expected value and standard deviation of the number of small aircraft that arrive during a 75-min period?
5. At a carnival cotton candy and hot dogs are two different prices. Adam buys one cotton candy and one hot dog for $5. Shannon buys 3 cotton candies and one hot dog for $11. How
much is one hot dog? How much is one cotton candy?
-2.3f+0.7f-12-3=
please help me on this
Answer:
-8/5f - 15
or
-1.6f - 15
Step-by-step explanation:
-2.3f + 0.7f - 12 - 3 =
-8/5f - 15
or
-1.6f - 15
Answer:
[tex] \sf \: -1.6f - 15 [/tex]
Step-by-step explanation:
Now we have to,
→ Simplify the given expression.
The expression is,
→ -2.3f + 0.7f - 12 - 3
Let's simplify the expression,
→ -2.3f + 0.7f - 12 - 3
→ (-2.3f + 0.7f) - 12 - 3
→ (-1.6f) + (-12 - 3)
→ -1.6f + (-15)
→ -1.6f - 15
Hence, the answer is -1.6f - 15.
Which value of n makes the equation true?
-1/2n=-8
O A. -16
OB. -4
O c. 4
O D. 16
Answer:
D. 16
Step-by-step explanation:
[tex]-\frac{1}{2} n=-8[/tex]
multiply by -2/1 on both sides
[tex](-2)-\frac{1}{2} n=-8(-2)[/tex]
simplify
[tex]n=16[/tex]
wement of the progress bar may be uneven because questions can be worth more or less (including zero) depes
What is the slope of the line that contains the points (2, 5) and (4, -3)?
Answer: 75
Step-by-step explanation:
7+5=75
One spring day, Aaron noted the time of day and the temperature, in degrees Fahrenheit. His findings are as follows: At 6 a.m., the temperature was 52° F. For the next 3 hours, the temperature rose 2° per hour. For the next 4 hours, it rose 1° per hour. The temperature then stayed steady until 6 p.m. For the next 2 hours, the temperature dropped 2° per hour. The temperature then dropped steadily until the temperature was 57° at midnight. On the set of axes below, graph Aaron's data.
Answer:
see below
Step-by-step explanation:
I am not able to draw graphs or diagrams. However, I can provide a description of how to graph Aaron's data on a set of axes.
On the horizontal axis, mark the hours of the day from 6 a.m. to midnight.On the vertical axis, mark the temperature from 50°F to 60°F, in increments of 1°F or 2°F.At 6 a.m., plot a point at the intersection of the hour and temperature values, which is (6, 52).From 6 a.m. to 9 a.m., the temperature rose 2°F per hour. This means that at 7 a.m., the temperature was 54°F, at 8 a.m., it was 56°F, and at 9 a.m., it was 58°F. Plot these points on the graph.From 9 a.m. to 1 p.m., the temperature rose 1°F per hour. This means that at 10 a.m., the temperature was 59°F, at 11 a.m., it was 60°F, at 12 p.m. (noon), it was 61°F, and at 1 p.m., it was 62°F. Plot these points on the graph.From 1 p.m. to 6 p.m., the temperature stayed steady. This means that the temperature remained at 62°F during this time period. Plot a horizontal line at 62°F on the graph.From 6 p.m. to 8 p.m., the temperature dropped 2°F per hour. This means that at 7 p.m., the temperature was 60°F, and at 8 p.m., it was 58°F. Plot these points on the graph.From 8 p.m. to midnight, the temperature dropped steadily. This means that at 9 p.m., the temperature was 57°F, and it continued to decrease until it reached 57°F at midnight. Plot these points on the graph to complete the graph of Aaron's data.Note.The horizontal axis could be marked with a scale of every hour, while the vertical axis could be marked in increments of 2°F to fit all the data.
Among all right circular cones with a slant height of , what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the outer edge of the base to the vertex
The radius and height that maximize the volume of the cone with a slant height of 12 are both 6√2.
To maximize the volume of a cone with a slant height of 12, we need to find the dimensions (radius and height) that give the maximum volume. Use this formula to get the volume of a cone:
[tex]V = 1/3 * \pi * r^2 * h[/tex]
where r and h represents radius ad height respectively.
From the problem statement, we know that the slant height s of the cone is 12. We can use the Pythagorean theorem to relate the height h, radius r, and slant height s:
[tex]s^2 = r^2 + h^2[/tex]
Substituting s = 12, we get:
[tex]12^2 = r^2 + h^2[/tex]
[tex]144 = r^2 + h^2[/tex]
We want to maximize the volume V, so we need to express it in terms of just one variable. We can use the equation relating r, h, and s:
[tex]s^2 = r^2 + h^2[/tex]
Solving for [tex]h^2[/tex], we get:
[tex]h^2 = s^2 - r^2[/tex]
Substituting s = 12, we get:
[tex]h^2 = 144 - r^2[/tex]
Now we can express the volume V as a function of just one variable, r:
[tex]V = 1/3 * \pi * r^2 * h[/tex]
[tex]V = 1/3 * \pi * r^2 * \sqrt{(144 - r^2)[/tex]
To find the maximum volume, we need to take the derivative of V with respect to r, set it equal to zero, and solve for r:
[tex]dV/dr = 1/3 * \pi * (2r * \sqrt{(144 - r^2)} - 2r^3 / \sqrt{(144 - r^2)})[/tex]
[tex]dV/dr = \pi r / 3 * (2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)})[/tex]
Setting dV/dr = 0, we get:
[tex]2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)} = 0[/tex]
[tex]2(144 - r^2) - 2r^2 = 0[/tex]
[tex]288 - 4r^2 = 0[/tex]
[tex]r^2 = 72[/tex]
So the radius that maximizes the volume is [tex]r = \sqrt{(72)} = 6\sqrt{(2)[/tex]. To find the height h, we can use the equation [tex]h^2 = 144 - r^2[/tex]:
[tex]h^2 = 144 - (6\sqrt{(2)})^2 = 72[/tex]
[tex]h = \sqrt{(72)} = 6\sqrt{(2)[/tex]
Therefore, the dimensions (radius and height) that maximize the volume of the cone with a slant height of 12 are [tex]r = 6\sqrt{(2)[/tex] and [tex]h = 6\sqrt{(2)[/tex].
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The complete question is:
What are the dimensions (radius and height) of all right circular cones with a slant height of 12 that maximize the cone's volume? The cone's slant height is the distance from the outer edge of the base to the vertex.
Please help!
Today, the temperature in Boston
is -1 degrees and it is expected to rise
3 degrees each day.
It's 6 degrees in San Francisco and is
expected to fall 1 degree every two
days.
After how many days will the
two temperatures be the same?
Equation 1:
Equation 2:
Answer:
Answer:
The temperature will be the same in two days
Step-by-step explanation:
We can set up two equations to represent the temperatures in Boston (B) and San Francisco (SF) after a certain number of days (d):
B = -1 + 3d
SF = 6 - (d/2)
To find the number of days when the two temperatures will be the same, we can set the equations equal to each other and solve for d:
-1 + 3d = 6 - (d/2)
Multiplying both sides by 2 to eliminate the fraction:
-2 + 6d = 12 - d
Combining like terms:
7d = 14
Dividing both sides by 7:
d = 2
Therefore, the two temperatures will be the same after 2 days.
The sum of my digits is equal to 5
My tens digit is not odd
( t - u ) = 1
Keep in mind there are two digits
The two digits are 3 and 2.
What are numbers?A number is a numerical unit of measurement and labeling in mathematics.
The natural numbers 1, 2, 3, 4, and so on are the first examples.
Number words are a linguistic way to express numbers.
Most commonly, specific numbers can be represented using symbols known as numerals; for instance, the numeral "5" stands in for the number five.
Basic numerals are frequently grouped in a numeral system, which is an ordered manner to represent any number, because only a small number of symbols can be learned.
The most widely used number system is the Hindu-Arabic number system, which enables the depiction of any number using a combination of 10 basic numerical symbols, or "digits."
According to our question-
3+2
=5
Hence, The two digits are 3 and 2.
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