I need someone to check my work not sure if I'm doing this right.

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 times the height, what should each dimension be? HINT: Let x represent one of the dimensions, then define the other dimensions in terms of x.

I got h=1.6 in
w=2in
l=2.4in
l+w=H=6
L=1.5H
1.5H+W+H=6
2.5W=6
W=6-2.5H
H=X
L=1.5X
V(X)=(1.5X)(X)(6-2.5X)
V(X)=9X^2-3.73X^3
V1(X)=18X-11.25X^2
X=1.6, (X=0)
V2(X)=18-2.25X
V2(16)=-18<0
X=16
H=1.6
W=2
L=2.4

L+W+H=6

L=1.5*H
2.5H+w=6

If H=x then W=6-2.5x and L=1.5x
v(x)=L*H*W= x(1.5x)(6-2.5x)= 9x^2-3.75x^3
V'(x)= 18x-11.25x^2
For stationary point; V'(x)=0 so x=0 reject or 18=11.25x so x=18/11.25= 1.6
check if it's a max: V''(x)=18-22.5x at x=1.6 V''(1.6)= -18<0 so yes
H=1.6, W=2, L= 2.4

YOUR ANSWER IS CORRECT. JUST IN LINE V(X)=9X^2-3.73X^3 you've made a mistake as it's 3.75 not 3.73

Thank you

To check your work, let's go through the steps:

1. Let x represent one of the dimensions.

2. The other dimensions can be defined in terms of x. Based on the information given, the length should be 1.5 times the height. Therefore, the length (L) would be equal to 1.5x.

3. The sum of the dimensions is given as 6 inches. So, the equation would be: L + x + x = 6.

4. Simplifying the equation: 1.5x + 2x = 6.

5. Combining like terms: 3.5x = 6.

6. Solving for x: x = 6 / 3.5.

7. Evaluating x: x ≈ 1.71 inches.

8. Now we can find the values of all three dimensions:

- Length (L): L = 1.5x ≈ 1.5 * 1.71 ≈ 2.57 inches.
- Width (W): W = x ≈ 1.71 inches.
- Height (H): H = x ≈ 1.71 inches.

So, the dimensions should be approximately:
- Length (L) ≈ 2.57 inches.
- Width (W) ≈ 1.71 inches.
- Height (H) ≈ 1.71 inches.

Your previous values were close, but there were some small errors in the calculations.

To check your work, let's calculate the sum of the dimensions and see if it equals 6 inches, and also check if the length is 1.5 times the height.

Sum of the dimensions:
h = 1.6 in
w = 2 in
l = 2.4 in

Sum of dimensions = h + w + l
= 1.6 + 2 + 2.4
= 6 inches

Great! The sum of the dimensions is indeed 6 inches.

Now let's check if the length is 1.5 times the height:
Length = l
Height = h

Length = 1.5 * Height
= 1.5 * 1.6
= 2.4 inches

Again, your calculation is correct. The length is indeed 1.5 times the height.

Therefore, based on your calculations:
Height = 1.6 inches
Width = 2 inches
Length = 2.4 inches.

Your answer is correct. Well done!