Find the volume of a rectangular box. The faces have areas of 15, 25, and 60.

factor the areas to see that

15 = 3*5
25 = 5*5
60 = 12*5

Hmm. No obvious integer solutions

15 = 6 * 5/2
25 = 5/2 * 10
60 = 10 * 6

v = 2.5 x 6 x 10 = 150

To find the volume of a rectangular box, you need to know the areas of its faces. In this case, you have three face areas: 15, 25, and 60.

Let's call the length, width, and height of the rectangular box L, W, and H respectively. We can use these values to form three equations using the given face areas.

Equation 1: L * W = 15
Equation 2: W * H = 25
Equation 3: L * H = 60

Now, we need to solve these equations to find the values of L, W, and H. We can start by rearranging the equations:

Equation 1: W = 15 / L
Equation 2: H = 25 / W
Equation 3: H = 60 / L

Since equations 2 and 3 both involve H, we can equate them:

25 / W = 60 / L

Cross-multiplying gives us:

25L = 60W

We can rearrange this equation to solve for W:

W = (25L) / 60

Substituting this value of W into equation 1:

(15 / L) * ((25L) / 60) = 15

Simplifying:

375L² = 900

Dividing both sides by 375:

L² = 900 / 375

L² = 2.4

Taking the square root of both sides:

L ≈ √2.4

Similarly, substituting the value of L into equation 3:

(60 / L) * ((25L) / 60) = 25

Simplifying:

25L = 25

L = 1

We have found the value of L as 1. Substituting this value into equation 1:

W = 15 / (1)

W = 15

Substituting the values of L and W into equation 2:

H = 25 / (15)

H ≈ 1.67

So, the values of L, W, and H are approximately 1, 15, and 1.67 respectively.

Finally, we can calculate the volume of the rectangular box using the formula:

Volume = Length * Width * Height

Volume = 1 * 15 * 1.67

Volume ≈ 25.05 cubic units

Therefore, the volume of the rectangular box is approximately 25.05 cubic units.