Question

The design of a digital box camera maximizes the volume while keeping the sum of the dimensions at 6 inches. If the length must be 1.5 time the height, what should each dimension be?

Hint: Let x represent one of the dimensions, and then define the other dimensions in terms of x.

Answer 1

Let x be the height, l be the length, and w be the width.
Can you find an equation to relate x and l?

Answer 2

idk I am extremely confused

Answer 3

We're told that the length is 1.5 times the height, so l=1.5x
What else are we told about the different lengths that we can make an equation out of?

Answer 4

the sum has to be 6 inches???

Answer 5

Yes. Using x, l, and w, what would that equation be?

Answer 6

idk....

Answer 7

The sum of the dimensions is length+width+height.
Since we know this has to add up to 6, we have:
x+l+w=6

Answer 8

ok

Answer 9

Since we know that l=1.5x, we can say that x+1.5x+w=6, then solve for w in terms of x.

Answer 10

how would you do that?

Answer 11

\[x+1.5x+w = 6 \\ 2.5x+w=6 \\ w=6-2.5x\]

Answer 12

Now we have length, width, and height all in terms of one variable. What would the volume be?
l= 1.5x
w= 6-2.5x
h= x

Answer 13

6.5?

Answer 14

\[(1.5x)(6-2.5x)(x) \\ 1.5x^2(6-2.5x) \\ 9x^2 - 3.75x^3\]

Do you know how to find the maximum of that?

Answer 15

no sorry

Answer 16

To find a relative max/min of a function, take the derivative and find the zeroes.

Answer 17

i have no idea what u r saying

Answer 18

What class is this for?

Answer 19

math

Answer 20

What level math? (Algebra, Geometry, Calculus?)

Answer 21

Algebra

Answer 22

pre algebra

Answer 23

Well, we got to our equation for volume, and we're trying to find the largest possible volume. I would recommend graphing the equation, then see where the greatest volume is. Other than that, I don't know how to solve this using pre-algebra math.