Advanced Placement (AP): What is the rate of change of the volume of the square prism at that point

What is the rate of change of the volume of the, №9425639, 28.01.2021 23:51

Mathematics

The side of the base of a square prism is increasing at a rate of 5 meters per second and the height of the prism as decreasing at a rate of 2 meters per second. at a certain instant, the base s side is 6 meters and the height is 7 meters. what is the rate of change of the volume of the prism at that instant fin cubic meters per second? a. 348 b. 492 c. -492 d. 318 the volume of a square pnsm with base side s and neignth is s² h.

Step-by-step answer

P Answered by Specialist

4

348

Step-by-step explanation:

348

Step-by-step explanation:

The volume of the square prisma is given by the following formula:

In which h is the height, and s is the side of the base.

Let's use implicit derivatives to solve this problem:

In this problem, we have that:

So the correct answer is:

348

Mathematics

The side of the base of a square prism is increasing at a rate of 5 meters per second and the height of the prism as decreasing at a rate of 2 meters per second. at a certain instant, the base s side is 6 meters and the height is 7 meters. what is the rate of change of the volume of the prism at that instant fin cubic meters per second? a. 348 b. 492 c. -492 d. 318 the volume of a square pnsm with base side s and neignth is s² h.

Step-by-step answer

P Answered by Master

4

348

Step-by-step explanation:

348

Step-by-step explanation:

The volume of the square prisma is given by the following formula:

In which h is the height, and s is the side of the base.

Let's use implicit derivatives to solve this problem:

In this problem, we have that:

So the correct answer is:

348

Mathematics

The side of the base of a square prism is decreasing at a rate of 7 km/min and the height of the prism is increasing at a rate of 10 km/min. At a certain instant, the base side is 4 km and the height is 9 km. What is the rate of change of the surface area of the prism at that instant

Step-by-step answer

P Answered by PhD

-204 km^2/min

Step-by-step explanation:

For base edge length s and height h, the surface area of the prism is ...

A = 2(s^2 +2sh) = 2s^2 +4sh

Then the rate of change of surface area is ...

A' = 4s·s' +4s'·h +4s·h' = 4s(s' +h') +4s'·h

Filling in the given values, we find the rate of change of area to be ...

A' = 4(4 km)(-7 km/min +10 km/min) +4(-7 km/min)(9 km)

A' = 4(12 km^2/min -63 km^2/min)

A' = -204 km^2/min

The area is decreasing at the rate of 204 square km per minute.

Mathematics

The length s(t)s(t)s, (, t, )of the side of the base of a square prism is decreasing at a rate of 777 kilometers per minute and the height h(t)h(t)h, (, t, )of the prism is increasing at a rate of 101010 kilometers per minute. At a certain instant t_0t 0 t, start subscript, 0, end subscript, the base s side is 444 kilometers and the height is 999 kilometers. What is the rate of change of the surface area A(t)A(t)A, (, t, )of the prism at that instant?

Step-by-step answer

P Answered by PhD

$\frac{dA_{s}}{dt} = -148\,\frac{km^{2}}{min}$

Step-by-step explanation:

The surface area of the square prism is obtained by using the following formula:

$A_{s} (t) = 4\cdot l(t)\cdot h(t) + 2\cdot [l(t)]^{2}$

The rate of change of the surface area can be found by deriving the function with respect to time:

$\frac{dA_{s}}{dt} = 4\cdot [h(t)\cdot \frac{dl}{dt} + l(t)\cdot \frac{dh}{dt}] + 2\cdot l(t)\cdot \frac{dl}{dt}$

Known variables are summarized below:

$h(t) = 9\,km$

$l(t) = 4\,km$

$\frac{dh}{dt} = 10\,\frac{km}{min}$

$\frac{dl}{dt} = -7\,\frac{km}{min}$

The rate of change is:

$\frac{dA_{s}}{dt} = 4\cdot [(9\,km)\cdot (-7\,\frac{km}{min} )+(4\,km)\cdot (10\,\frac{km}{min} )] + 2\cdot (4\,km)\cdot (-7\,\frac{km}{min} )$

$\frac{dA_{s}}{dt} = -148\,\frac{km^{2}}{min}$

Physics

The side of the base of a square prism is increasing at a rate of 555 meters per second and the height of the prism is decreasing at a rate of 222 meters per second. at a certain instant, the base s side is 666 meters and the height is 777 meters. what is the rate of change of the volume of the prism at that instant (in cubic meters per second)?

Step-by-step answer

P Answered by PhD

Answer options:
A. -348
B. 492
C. -492
D. 348

Answer: D. 348.

Explanation:
The volume of the square prisma is given by the following formula:
V=s^2*h.
In which h is the height, and s is the side of the base.
Let's use implicit derivatives to solve this problem:
dV/dt=2sh*ds/dt+s^2*dh/dt.
In this problem, we have that:
ds/dt=5, dh/dt=-2, h=7, s=6.
So
dV/dt=2sh*ds/dt+s^2*dh/dt;
dV/dt=2*6*7*5+(6)^2*(-2)=348 m/s^3.
So the correct answer is:
D. 348.

Mathematics

The side of the base of a square prism is decreasing at a rate of 7 km/min and the height of the prism is increasing at a rate of 10 km/min. At a certain instant, the base side is 4 km and the height is 9 km. What is the rate of change of the surface area of the prism at that instant

Step-by-step answer

P Answered by PhD

-204 km^2/min

Step-by-step explanation:

For base edge length s and height h, the surface area of the prism is ...

A = 2(s^2 +2sh) = 2s^2 +4sh

Then the rate of change of surface area is ...

A' = 4s·s' +4s'·h +4s·h' = 4s(s' +h') +4s'·h

Filling in the given values, we find the rate of change of area to be ...

A' = 4(4 km)(-7 km/min +10 km/min) +4(-7 km/min)(9 km)

A' = 4(12 km^2/min -63 km^2/min)

A' = -204 km^2/min

The area is decreasing at the rate of 204 square km per minute.

Mathematics

The length s(t)s(t)s, (, t, )of the side of the base of a square prism is decreasing at a rate of 777 kilometers per minute and the height h(t)h(t)h, (, t, )of the prism is increasing at a rate of 101010 kilometers per minute. At a certain instant t_0t 0 t, start subscript, 0, end subscript, the base s side is 444 kilometers and the height is 999 kilometers. What is the rate of change of the surface area A(t)A(t)A, (, t, )of the prism at that instant?

Step-by-step answer

P Answered by PhD

$\frac{dA_{s}}{dt} = -148\,\frac{km^{2}}{min}$

Step-by-step explanation:

The surface area of the square prism is obtained by using the following formula:

$A_{s} (t) = 4\cdot l(t)\cdot h(t) + 2\cdot [l(t)]^{2}$

The rate of change of the surface area can be found by deriving the function with respect to time:

$\frac{dA_{s}}{dt} = 4\cdot [h(t)\cdot \frac{dl}{dt} + l(t)\cdot \frac{dh}{dt}] + 2\cdot l(t)\cdot \frac{dl}{dt}$

Known variables are summarized below:

$h(t) = 9\,km$

$l(t) = 4\,km$

$\frac{dh}{dt} = 10\,\frac{km}{min}$

$\frac{dl}{dt} = -7\,\frac{km}{min}$

The rate of change is:

$\frac{dA_{s}}{dt} = 4\cdot [(9\,km)\cdot (-7\,\frac{km}{min} )+(4\,km)\cdot (10\,\frac{km}{min} )] + 2\cdot (4\,km)\cdot (-7\,\frac{km}{min} )$

$\frac{dA_{s}}{dt} = -148\,\frac{km^{2}}{min}$

Mathematics

The side of the base of a square prism is increasing at a rate of 5 meters per second and the height of the prism as decreasing at a rate of 2 meters per second. at a certain instant, the base s side is 6 meters and the height is 7 meters. what is the rate of change of the volume of the prism at that instant fin cubic meters per second? a. -348 b. 492 c. -492 d. 348

Step-by-step answer

P Answered by PhD

D. 348

Step-by-step explanation:

The volume of the square prisma is given by the following formula:

$V = s^{2}h$

In which h is the height, and s is the side of the base.

Let's use implicit derivatives to solve this problem:

$\frac{dV}{dt} = 2sh\frac{ds}{dt} + s^{2}\frac{dh}{dt}$

In this problem, we have that:

$\frac{ds}{dt} = 5, \frac{dh}{dt} = -2, h = 7, s = 6$

So

$\frac{dV}{dt} = 2sh\frac{ds}{dt} + s^{2}\frac{dh}{dt}$

$\frac{dV}{dt} = 2*6*7*5 + (6)^{2}*(-2) = 348$

So the correct answer is:

D. 348

Mathematics

A right square prism has a volume of 75 cubic inches. The prism is enlarged so its height is increased by a factor of 5, but the other dimensions do not change. What is the new volume? A. B. C. D.

Step-by-step answer

P Answered by Specialist

The new volume is 375 cubic inches.

Step-by-step explanation:

Volume of a right square prism:

The volume of a right square prism is given by:

V = a^2h

In which a is the length of the edge and h is the height.

A right square prism has a volume of 75 cubic inches.

This means that:

a^2h = 75

The prism is enlarged so its height is increased by a factor of 5

Edge is the same, height is 5 times. So

a^2(5h) = 5(a^2h) = 5*75 = 375

The new volume is 375 cubic inches.

Mathematics

Select Transform to change the drawing of the figure. The previous triangular prism had a surface area of 288 square units. What happens to the surface area of your prism when you double all four measurements? The surface area doubles. O The surface area triples. The surface area increases by 4 times. The surface area increases by 8 times. 12 in 20 in. 16 in. 20 in. Transform

Step-by-step answer

P Answered by Specialist

41

C- The surface area is increased by 4 times

Step-by-step explanation:

What is the rate of change of the volume of the square prism at that point

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