An architect is planning a building in the shape of a rectangular prism. The length of the building is 20 feet more than its width and the height is 10 feet more than twice its width The inside volume of the building is 2,000,000 cubic feet. A model of the building is constructed using the scale 10 feet to 2 inches. Find the dimensions of the model in inches.
I am having trouble with this story problem.
If the width is w, then we have
w(w+20)(2w+10) = 2000000
solve that for w, and you have w ≈ 92.017
Just for grins, say w=92 (then the volume is 92x112x194 = 1,998,976)
Anyway, just multiply the dimensions by 2"/10' = 1/60 to get the dimensions in inches.
I guess, since it won't come out even, you might as well use w=92.017 ft
To find the dimensions of the model in inches, we need to first determine the dimensions of the actual building in feet. Then, we can convert these measurements to inches using the scale provided.
Let's solve the problem step by step:
1. Let's denote the width of the building as "x" feet. Therefore, the length of the building is (x + 20) feet, and the height is (2x + 10) feet.
2. The volume of a rectangular prism is calculated by multiplying its length, width, and height. In this case, the inside volume of the building is given as 2,000,000 cubic feet, so we have the equation:
(x + 20) * x * (2x + 10) = 2,000,000
3. Simplifying the equation:
(x^2 + 20x) * (2x + 10) = 2,000,000
2x^3 + 40x^2 + 10x^2 + 200x = 2,000,000
2x^3 + 50x^2 + 200x - 2,000,000 = 0
4. Now we need to solve this cubic equation. Unfortunately, there is no simple method to solve cubic equations. We can either approximate the solution using numerical methods or use a graphing calculator to find the roots of the equation.
Assuming you have access to a graphing calculator or software, enter the equation (2x^3 + 50x^2 + 200x - 2,000,000 = 0) and find the roots. The value of "x" will give us the width of the building in feet.
5. Once we have the width of the building in feet, we can calculate the length and height by plugging in the value of "x" into the expressions:
Length = x + 20
Height = 2x + 10
6. To convert the dimensions from feet to inches, we need to use the provided scale of 10 feet to 2 inches. Convert each measurement from feet to inches by multiplying by 12 (since there are 12 inches in a foot).
Width (in inches) = Width (in feet) * 12 * 10
Length (in inches) = Length (in feet) * 12 * 10
Height (in inches) = Height (in feet) * 12 * 10
By following these steps, you should be able to find the dimensions of the model of the building in inches.