I have a question: My textbook says "The surface of a right prism is 224 sq. feet, and the length of a side of the square base is one-third the height. What are the dimensions of the prism?" How do I find out the dimensions when I don't know the height or the length of the sides? Please help. I have been working like crazy.
surface area = 2x^2 + 4xh
where x is the length of the side and h is the height.
let x= 1/3h
Then 224 = (2/9)h^2 + (4/3)h^2
solve it h = 12
then x = 1/3 h = 4
So the length is 4 and height is 12
4/3
To find the dimensions of the right prism, we first need to understand the relationship between the surface area, base side length, and height.
Let's set up the problem step by step:
1. Let's denote the base side length as "x" and the height as "h".
2. The surface area of the right prism is given as 224 square feet. The surface area of a prism is calculated by adding up the areas of all its faces. In this case, we have two congruent rectangular faces (the bases) and four rectangular faces (the lateral faces). Since the prism is a right prism, the lateral faces are perpendicular to the bases.
3. The area of each base is x * x = x^2 square feet. Since we have two bases, the total area of the bases is 2 * x^2 square feet.
4. The combined area of the lateral faces can be calculated by multiplying the perimeter of the base with the height of the prism. Since the base is a square, the perimeter is given by 4 * x. Therefore, the area of the four lateral faces is 4 * x * h square feet.
5. Adding up the areas of the bases and the lateral faces, we get the total surface area: 2 * x^2 + 4 * x * h = 224 square feet.
Now we have a system of equations:
2 * x^2 + 4 * x * h = 224, and
x = h/3.
We can solve this system of equations to find the values of x (base side length) and h (height).
To do that, let's substitute the value of x from the second equation into the first equation:
2 * (h/3)^2 + 4 * (h/3) * h = 224.
Simplifying, we have:
2 * (h^2/9) + 4 * (h/3) * h = 224,
(h^2/9) + 2 * h^2 = 224.
Multiplying through by 9 to clear the fraction, we get:
h^2 + 18 * h^2 = 2016,
19 * h^2 = 2016,
h^2 = 2016/19,
h ≈ 8.12 feet.
Now that we have the value of h (approximately 8.12 feet), we can substitute it back into the second equation to find the value of x:
x = h/3,
x ≈ 8.12/3,
x ≈ 2.71 feet.
Therefore, the dimensions of the right prism are approximately: base side length x ≈ 2.71 feet and height h ≈ 8.12 feet.