Ask Me Help Desk - 1 Precalculus Optimization Q and 1 Geometric Sequence Q

I have tried both of these, but cannot seem to get answers. I get at least one negative dimension for the first geometric sequence problem, which would be impossible and render the question having no solution. However, I am pretty sure there is indeed a required solution. For the next question I simply do not know how to draw the correct diagram and how to derive any equations from it to optimize so I am pretty much lost. Help is very much appreciated.

1. The volume of a certain rectangular solid is 64 cubic centimeters and its total surface area is 384 square centimeters. Its three dimensions are in geometric progression. What is the sum of the lengths of all the edges of this solid?

2. Town A is 16 miles from a straight river and Town B is 7 miles from the same river. The distance from Town A to Town B is 15 miles. A pumping station is to be built along the river to supply water to both towns. Where should the pumping station be built so that the sum of the distances from the pumping station to the towns is a minimum? Include a diagram and answer to the nearest hundredth. Assume Town A is west of Town B.

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2. Town A is 16 miles from a straight river and Town B is 7 miles from the same river. The distance from Town A to Town B is 15 miles. A pumping station is to be built along the river to supply water to both towns. Where should the pumping station be built so that the sum of the distances from the pumping station to the towns is a minimum? Include a diagram and answer to the nearest hundredth. Assume Town A is west of Town B.

The point we must find is at point P on the diagram.

The distance along the shore between the towns is 12 miles. Because $\sqrt{15^{2}-(16-7)^{2}}=12$

We must find the length of x by using Pythagoras.

The sum of the distances is given by:

$\sqrt{49+(12-x)^{2}}+\sqrt{256+x^{2}}=D$

This is what must be minimized. Can you continue?

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1. The volume of a certain rectangular solid is 64 cubic centimeters and its total surface area is 384 square centimeters. Its three dimensions are in geometric progression. What is the sum of the lengths of all the edges of this solid?

Each term of a geometric progession is given by $a_{n}=a_{1}r^{n-1}$

Let's allow one side to be $x=a_{1}$

Then side y will be $a_{2}=xr^{1}=y$

And side z will be $a_{3}=xr^{2}=z$

The volume is $xyz=a_{1}(a_{1}r)(a_{1}r^{2})=64$

The surface area is:

$2xz+2yz+2xy=2a_{1}(a_{1}r^{2})+2(a_{1}r)(a_{1}r^{2 })+2a_{1}(a_{1}r)=364$

Solve for $a_{1}$ and r. Then, since $x=a_{1}$ , you can find y and z.

Thanks for your help so far. For the optimization problem, after you get the equation to be optimized, you just graph it and get 8.35 as the minimum for x. But for the other question, I get 8a^2-332a+128 as the equation for a1. However, this leads to 2 irrational answers for a1. Am I doing something wrong here?

For #2, the solutions I got were not nice integers, either. Looks like you're on the right track.

K I got 192 as the answer for #2. That was quite a long question if you figure everything out in radical form, probably took me 10 minutes to solve for all of the sides from the equation.