Expert: Paul Klarreich - 5/8/2007

Question
1- The three dimension of a box are increasing at the rate of 5 cm/min, 7 cm/min, and 2 cm/min. At what rate is the volume increasing at the moment when the box is a cube with edge 10 cm? At what rate is the surface area increasing?

2- The base of a triangle is increasing at the rate of ¨icm/min, while the altitude is decreasing at the same rate. At what rate is the atea changing when (a) the base is 10 cm and the altitude is 6 cm? (b) the base is 6 cm and the altitude is 10 cm?

3- The area of a circle is increasing at te rate of ¨icm/min. At what rate is the radius increasing when the atea is 4 ¨icm?

Answer
Questioner:   Antonio Tando
Category:  Calculus
 
Question:  1- The three dimension of a box are increasing at the rate of 5 cm/min, 7 cm/min, and 2 cm/min. At what rate is the volume increasing at the moment when the box is a cube with edge 10 cm? At what rate is the surface area increasing?

2- The base of a triangle is increasing at the rate of ¨icm/min, while the altitude is decreasing at the same rate. At what rate is the atea changing when (a) the base is 10 cm and the altitude is 6 cm? (b) the base is 6 cm and the altitude is 10 cm?

3- The area of a circle is increasing at te rate of ¨icm/min. At what rate is the radius increasing when the atea is 4 ¨icm?
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Hi, Antonio,

I am sorry, but some of your information got messed up in the translation.  In the future, DO NOT use any special characters.  They get lost.  I will make up my own numbers.
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Is this your first attempt at R-R problems?  If so, the scheme is something like this:

1. Identify the variables in the problem -- the things that change.  Give them names.

2. Write their rates of change as derivatives WITH RESPECT TO time.  Note which are known and which is to be found.

3. Determine a relationship (yes, it is called 'related rates' for a reason) between the variables.  Use a diagram, use your life experience, use your general knowledge and brilliance, do whatever you have to.  This is the key step.

4. Now differentiate implicitly, then substitute the known quantities and rates, and solve for the unknown rate.
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Q1:  the variables are L,W,H, and V.

V = LWH, and each is a function of t.

dV
-- = WH dL/dt + LH dW/dt + LW dh/t
dt

If the box is a cube, we use:

W = H = L = 10
dL/dt = 5
dW/dt = 7
dH/dt = 2

Substitute:

dV
-- = 100(5) + 100(2) + 100(7) = 1400
dt
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2- The base of a triangle is increasing at the rate of ¨icm/min, while the altitude (height) is decreasing at the same rate. At what rate is the atea changing when (a) the base is 10 cm and the altitude is 6 cm? (b) the base is 6 cm and the altitude is 10 cm?

I will use 'r' for the rate at which the base is increasing.  Whatever number you have, just put it in.


A = bh/2

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

a) Use  b = 10,  h = 6
db/dt = r, and dh/dt = -r,  and you have:

dA/dt = 1/2( -10r + 6r)
= 1/2(-4r) = - 2r.

b) Just put the values in as we did for part a.  I'll leave that to you.

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3- The area of a circle is increasing at te rate of ¨icm/min. At what rate is the radius increasing when the area is 4 ¨icm?

Your rates and area got messed up.  I think you tried to put in the symbol for pi, so I'll use that.  I'll assume your problem said:

3- The area of a circle is increasing at te rate of pi cm^2/min. At what rate is the radius increasing when the area is 4pi cm^2?

A = pi r^2

dA/dt = 2 pi r dr/dt

Now use r = 2.  Why r = 2?  Because you said  A = 4pi = pi r^2.
Also, dA/dt = pi.

pi = 2 pi(2) dr/dt
1 = 4 dr/dt
dr/dt = 1/4 cm/sec.