Answer:
A)
[tex]P=2x+2\pi r[/tex]
B)
[tex]x=\frac{1}{2}(P-2\pi r)[/tex]
C)
[tex]x\approx173\text{ feet}[/tex]
Step-by-step explanation:
So, the track is composed of a rectangle and two semicircles.
The side length of the rectangle is x, and the radius of the semicircles is r.
Part A)
The perimeter is simply the length around the figure.
We can see that the perimeter will be equivalent to two times x plus two times the circumference of the semi-circles.
So, let's figure out the circumference of the semi-circles. The circumference of a normal circle is:
[tex]C_{\text{normal}}=2\pi r[/tex]
A semi-circle is half of that. So, we will divide by 2:
[tex]C_{\text{ semi}}=\pi r[/tex]
We are excluding the diameter from this.
So, our perimeter will be two times x and two times the circumference of a semi-circle (since there are two semi-circles). In other words:
[tex]P=2(x)+2(\pi r)[/tex]
Simplify:
[tex]P=2x+2\pi r[/tex]
Part B)
So we acquired the formula:
[tex]P=2x+2\pi r[/tex]
To solve for x, first, subtract 2Ï€r from both sides. So:
[tex](P)-2\pi r=(2x+2\pi r)-2\pi r[/tex]
The right side cancels. So:
[tex]P-2\pi r=2x[/tex]
Multiply both sides by 1/2:
[tex]\frac{1}{2}(P-2\pi r)=\frac{1}{2}(2x)[/tex]
The right side cancels. So:
[tex]\frac{1}{2}(P-2\pi r)=x[/tex]
Flip:
[tex]x=\frac{1}{2}(P-2\pi r)[/tex]
Part C)
So we want to find x when the perimeter is 660 and r is 50 feet. So, substitute them into our formula. We have:
[tex]x=\frac{1}{2}(P-2\pi r)[/tex]
Substitute 660 for P and 50 for r:
[tex]x=\frac{1}{2}(660-2\pi (50))[/tex]
Multiply:
[tex]x=\frac{1}{2}(660-100\pi )[/tex]
Distribute:
[tex]x=330-50\pi[/tex]
Use a calculator. So:
[tex]x\approx173\text{ feet}[/tex]
And we're done!
