mathematical models(precalculus) problems

**^_^Engineer_Adam^_^** · November 20th 2006, 03:20 AM

Hello i Need Help on mathematical models

11. The surface area of a sphere is a function of its radius. If $r$ centimeters is the radius of a sphere and [tex]A(r)[tex] square centimeters is the surface area, then $A(r) = 4 \pi r^2$ . Suppose a balloon maintains the shape of a sphere as it is being inflated so that the radius is changing at a constant rate of 3 centimeters per second. If $f(t)$ centimeters is the radius of the balloon after $t$ seconds, do the following:
a.) Compute $(A \circ f)(t)$ and interpret your result.
b.) Find the surface area of the balloon after 4 seconds.

14. A rectangular garden is to be fenced off with 100ft of fencing material. a.) Find the mathematical model expressing the area of the garden as a function of its length. b.) what is the domain of your function in part a?
c.) By plotting on your a graphics calculator the graph of your function in part (a), estimate to the nearest foot the dimensions of the largest rectangular garden that can be fenced off with the 100ft of material

Thank you very much!!!

**CaptainBlack** · November 20th 2006, 03:34 AM

Originally Posted by ^_^Engineer_Adam^_^

Hello i Need Help on mathematical models

11. The surface area of a sphere is a function of its radius. If $r$ centimeters is the radius of a sphere and $A(r)$ square centimeters is the surface area, then $A(r) = 4 \pi r^2$ . Suppose a balloon maintains the shape of a sphere as it is being inflated so that the radius is changing at a constant rate of 3 centimeters per second. If $f(t)$ centimeters is the radius of the balloon after $t$ seconds, do the following:
a.) Compute $(A \circ f)(t)$ and interpret your result.

As the radius is increasing at $3 cm/s$ the radius at time $t$ is:

$<br /> f(t)=r_0+3t<br />$

where $r_0$ is the radius at $t=0$ .

Then $(A \circ f)(t)=4 \pi (r_0+3t)^2$ .

b.) Find the surface area of the balloon after 4 seconds.

Then the surface area at $t=4$ is:

$(A \circ f)(4)=4 \pi (r_0+12)^2$

(it may be that you are supposed to assume $r_0=0$ but
it does not say so I won't).

RonL

**CaptainBlack** · November 20th 2006, 03:54 AM

Originally Posted by ^_^Engineer_Adam^_^

14. A rectangular garden is to be fenced off with 100ft of fencing material. a.) Find the mathematical model expressing the area of the garden as a function of its length. b.) what is the domain of your function in part a?

If the length of the garden is $lft$ , then as we fence it with $100ft$ of fence its width is $(100-2l)/2 =50-l ft$ so
its area is:

$<br /> A(l)=l (50-l)<br />$

The domain is $[0,50]$ , as the length cannot be less than $0ft$ or greater than $50ft$ .

c.) By plotting on your a graphics calculator the graph of your function in part (a), estimate to the nearest foot the dimensions of the largest rectangular garden that can be fenced off with the 100ft of material

See attachment

RonL

**^_^Engineer_Adam^_^** · November 22nd 2006, 03:53 AM

Hello,

... could you please help me understand number 11?
the part where i dont understand is $r_o$ in $f(t) = r_0 +3t?$ how did you get it ... how did it specify a time equal to zero?
Please help

thanks again!

**CaptainBlack** · November 22nd 2006, 08:34 PM

Originally Posted by ^_^Engineer_Adam^_^

Hello,

... could you please help me understand number 11?
the part where i dont understand is $r_o$ in $f(t) = r_0 +3t?$ how did you get it ... how did it specify a time equal to zero?
Please help

thanks again!

What it said was that the radius was increasing at at rate of $3\ cm/s$ , which means that if it starts with a radius (which we are not told) of $r_0$ at $t=0$ , then at $t$ its radius must be $r_0+3t$ , as in $t\ s$ the radius will have increased by $3t\ cm$ .

Now the existance of an initial radius is based on personal experience with baloons. There is some minimum radius where you have put just sufficient air into them so that they are more-or-less spherical, but the material is not stretched, we can take this as corresponding to time $0$ and radius $r_0$ . It makes little sense to assume that at $t=0$ the radius is $0$ as the idea of a baloon with a radius of $0$ is silly we would be outside the region of validity of the model.

RonL

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