Answer:
L: 6
W: 2
H: 20
Explanation:
First, we need to know the expression to calculate the volume of rectangular prism which is:
V = L * W * H
The only thing we know is that the volume is 240 in³ and that the width is 4in less than length, the height is 2 in more than 3 times length. So, everything is set in function of the length, therefore, we'll call L = x and then solve for x
L = x
W = x - 4
H = 3x + 2
Let's replace the data in the formula of volume:
240 = x * (x - 4) * (3x + 2)
240 = (x² - 4x)(3x + 2)
Solving this we have:
240 = 3x³ - 10x² - 8x
Rearranging the equation:
3x³ - 10x² - 8x - 240 = 0
Now, in order to solve this, we need to factorize the left side:
3x³ - 18x² + 8x² - 48x + 40x - 240 = 0
3x²(x - 6) + 8x(x - 6) + 40(x - 6) = 0
(x - 6)(3x² + 8x + 40) = 0
Now that we have this, we can solve the value of x.
x1 -----> x - 6 = 0 ---> x = 6
x2 ---> (3x² + 8x + 40) = 0
Δ = 8² - 4*3*40 = 64 - 480 = -416
This value of -416 means that it's not possible to get a real number as x (only the imaginary numbers), therefore the value of x should be 6
x = L = 6
With this value we can calculate W and H:
H = 6 - 2 = 4
W = (3*6) + 2 = 20