Question

You are building a pair of ramps for a loading platform. the left ramp is twice as long as the right ramp. if 150 cubic feet of concrete are used to build the ramps, what are the dimensions of each ramp?

Answer 1

From the given figure, the length of the entire structure is 21x + 6.

Given that the length of the platform is 3x, thus the sum of the lengths of the left and the right ramp is given by 21x + 6 - 3x = 18x + 6.

Given that the left ramp is twice as long as the right ramp, this means that the ratio of the length of the left ramp to that of the right ramp is 2 : 1.

Thus, the length of the left ramp is 2(18x + 6) / 3 = 2(6x + 2) = 12x + 4 and the length of the right ramp is 6x + 2.

The volume of the right ramp is $\frac{1}{2}x(12x+4)(3x)=18x^3+6x^2$ and the volume of the right ramp is $\frac{1}{2}x(6x+2)(3x)=9x^3+3x^2$.

Given that 150 cubic feet of concrete are used to build the ramps, thus

$18x^3+6x^2+9x^3+3x^2=150 \\ \\ \Rightarrow27x^3+9x^2-150=0 \\ \\ \Rightarrow9x^3+3x^2-50=0 \\ \\ \Rightarrow x\approx1.67$

Thus, the length of the left ramp is 12(1.67) + 4 = 24.04 and the length of the right ramp is 6(1.67) + 2 = 12.02

The height of both ramps is 1.67 while the width of both ramps is 3(1.67) = 5.01.