We can find the dimensions of the ramps in three steps.
First, we will express the length of each ramp in terms of x.
Next, we will express the total volume of the ramps in terms of x.
We will find x and the dimensions of the ramps.
Length of the Ramps
Let's introduce the variables y and z for the length of the ramps. We are given that the left ramp is twice as long as the right ramp, so y=2z.
The diagram gives the total length of the two ramps and the opening.
This allows us to set up and solve an equation for z.
Since y=2z, this also gives the length of the left ramp.
2
&Length of right ramp:&& z=6x+2
&Length of left ramp:&& y=2z=12x+4
Volume of the Ramps
Let's summarize the dimensions of the ramps in terms of x.
Left Ramp
Right Ramp
Length
12x+4
6x+2
Width
3x
3x
Height
x
x
The volume of each ramp is half the product of its length, width, and height.
Let's write and expand the expression of the volume of the left ramp first.
According to the Rational Root Theorem, all rational solutions of this equation are quotients of a factor of the constant term, 150, and a factor of the leading coefficient, 27.
Let's list these factors.
Even if we only consider the positive values since x represents a length, these are a lot of possibilities to try.
Let's use instead a calculator to draw the graph and find the x-intercept.
We begin by pushing the Y= button and typing the equation in the first row.
To see the graph, you will need to adjust the window.
Push WINDOW, change the settings, and push GRAPH.
To find the x-intercept of the graph, push 2nd and TRACE and choose zero from the menu.
The calculator will prompt you to choose a left and right bound and to provide the calculator with a best guess of where the zero might be.
The calculator gives that the approximate value of the solution is 1.6666667.
Let's compare this to the list of factors of the constant term and the leading coefficient.
We can see that it is close to 53, which is a possible solution according to the Rational Root Theorem.
Let's check this value.
