Answer: 60(30) + 16h) = 79(15h)
Let's call the height of the rectangular prism "h".
The surface area of a rectangular prism is given by the formula:
SA = 2lw + 2lh + 2wh
The volume of a rectangular prism is given by the formula:
V = lwh
We're given that the ratio of the surface area to the volume is 79:60. So, we can write:
SA/V = 79/60
Substituting the values for the length, width, and height of the prism, we get:
(2(5)(3) + 2(5)(h) + 2(3)(h)) / (5)(3)(h) = 79/60
Simplifying the left-hand side, we get:
(30 + 10h + 6h) / 15h = 79/60
Combining like terms on the numerator, we get:
(30 + 16h) / 15h = 79/60
Cross-multiplying, we get:
60(30 + 16h) = 79(15h)
Expanding and simplifying, we get:
1800 + 960h = 1185h
Subtracting 960h from both sides, we get:
1800 = 225h
Dividing both sides by 225, we get:
h = 8
So, the height of the rectangular prism is 8 inches.
To solve for the height using a proportion, we can set up the following proportion:
(2lw + 2lh + 2wh) / lwh = 79/60
Substituting the given values, we get:
(2(5)(3) + 2(5)(h) + 2(3)(h)) / (5)(3)(h) = 79/60
Simplifying the left-hand side, we get:
(30 + 10h + 6h) / 15h = 79/60
Cross-multiplying, we get:
60(30 + 16h) = 79(15h)
We can then solve for h using algebraic manipulation as shown earlier.