Related Rates (rectangular prism)

The conversation is discussing the concept of related rates, where we are given the rates at which different variables are changing and we are asked to find the rate of change of another variable. In this case, the given rates are the increase in length, width, and height of a rectangular prism, and we are asked to find the rate of change of its volume when the dimensions are specific values.The solution involves setting up an equation for volume using the given rates and dimensions, and then finding the derivative of that equation to solve for dV/dt. The key is to use the product rule to find the derivative of the volume equation, and then substitute in the given values to find the final answer.
  • #1
bdraycott
3
0

Homework Statement



A rectangular prism has its length increasing by 12 cm/min, its width increasing by 4 cm/min and its height increasing by 2 cm/min. How fast is it's volume changing when the dimensions are 200 cm in length, 50 cm in width and 30 cm in height?


Homework Equations





The Attempt at a Solution



I am reworking a few things for an online calculus course that I am about to finish. This questions seems to have me baffled though. I have worked out the equations to arrive at dv/dt, however when I try to complete ( by attempting to arrive at (t) or use the quadratic equations the values I arrive at are always negative?

Here we go,

dL/dt= 12cm/min dW/dt=4cm/min dH/dt=3cm/min

How fast is the change occurring when∶ L=200 W=50 H=30

V=LWH

V=(200+12t)(50+4t)(30+3t)

dv/dt=10,000+800t+600t+48t^2 (30+3t)

=300000+20000t+24000t+1600t+18000t+1200t^2+1440t^2+96t^3

= 96t^3+ 2648t^2+63600t+300000

=2〖(96t)〗^2+2(2648t)+63600

=192t^2+5296t+63600

∴at t=2cm/min

This is where I get lost, I have tried using v(t)=x(t)y(t)z(t) and keep running into (-) values.Hopefully someone can shed a little light on this one for me.
 
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  • #2
Aren't you trying to find the volume when t = 0? So wouldn't that mean the volume is changing at a rate of 63600 cm^3/min? (Assuming your calculations are correct)
 
  • #3
bdraycott said:

Homework Statement



A rectangular prism has its length increasing by 12 cm/min, its width increasing by 4 cm/min and its height increasing by 2 cm/min. How fast is it's volume changing when the dimensions are 200 cm in length, 50 cm in width and 30 cm in height?


Homework Equations





The Attempt at a Solution



I am reworking a few things for an online calculus course that I am about to finish. This questions seems to have me baffled though. I have worked out the equations to arrive at dv/dt, however when I try to complete ( by attempting to arrive at (t) or use the quadratic equations the values I arrive at are always negative?

Here we go,

dL/dt= 12cm/min dW/dt=4cm/min dH/dt=3cm/min

How fast is the change occurring when∶ L=200 W=50 H=30

V=LWH

V=(200+12t)(50+4t)(30+3t)

dv/dt=10,000+800t+600t+48t^2 (30+3t)

=300000+20000t+24000t+1600t+18000t+1200t^2+1440t^2+96t^3

= 96t^3+ 2648t^2+63600t+300000

=2〖(96t)〗^2+2(2648t)+63600

=192t^2+5296t+63600

∴at t=2cm/min

This is where I get lost, I have tried using v(t)=x(t)y(t)z(t) and keep running into (-) values.Hopefully someone can shed a little light on this one for me.

Relevant Equations:

V (prism) = Bh = LWH

How to Tackle The Problem:

Whenever starting any related rates problem, always 1. )STATE your givens.

You already did it--but here's to summarize:

dL/dt = 12 cm/min
dw/dt = 4 cm/min
dh/dt = 2 cm/min

W = 50
L = 200
H = 30

Next 2.) Find out what it's ASKING.


"How fast is it's volume changing"

Ding ding ding! Basically, in English, it's asking you to solve for dV/dt.

Since V = lwh

Find dV/dt ... don't do any substitutions yet. After you've solved for dV/dt THEN 3.) substitute your givens in the equation.
I got an answer of 62,000 cm^3 / min . Not too far from moemoney's answer.
 
  • #4
carlodelmundo's is correct and is far simpler.

You just have to remember the multiplication identity for taking derivatives of the product of 3 variables. This will save you time rather than having to expand everything out.

Y = abc ==> Y' = a'bc + ab'c + abc' if I remember correctly.
 
  • #5
Yep that's correct, just an extension of the product rule.
 

1. What is a rectangular prism?

A rectangular prism is a three-dimensional shape with six rectangular faces, where each face is perpendicular to the adjacent faces. It is also known as a cuboid.

2. How do you find the surface area of a rectangular prism?

To find the surface area of a rectangular prism, you need to add the area of all six faces. The formula for surface area is A = 2(lw + lh + wh), where l, w, and h are the length, width, and height of the prism, respectively.

3. What is the formula for volume of a rectangular prism?

The formula for volume of a rectangular prism is V = lwh, where l, w, and h are the length, width, and height of the prism, respectively. This formula can also be written as V = Bh, where B is the area of the base and h is the height.

4. How do related rates apply to a rectangular prism?

Related rates refer to the rate at which one variable changes with respect to another variable. In the case of a rectangular prism, related rates can be used to determine how the volume or surface area of the prism changes as its dimensions change.

5. Can you give an example of a related rates problem involving a rectangular prism?

One example of a related rates problem involving a rectangular prism is as follows: A rectangular prism has a fixed volume of 100 cubic centimeters. If the height is decreasing at a rate of 2 cm/s and the length is increasing at a rate of 3 cm/s, what is the rate of change of the width? This problem can be solved using the formula for volume and related rates to find the rate of change of the width.

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