Question

the figure illustrates the apparatus for a tightrope walker. two poles are set y = 60 feet apart, but the point of attachment p for the rope is yet to be determined. (a) express the length l of the rope as a function of the distance x from p to the ground. l(x)= (b) if the total walk is to be 90 feet, determine the distance from p to the ground. (round your answer to one decimal place.) ft

Explanation:

Step1: Apply Pythagorean theorem

We have a right - triangle with horizontal side $y = 60$ and vertical side $(x - 2)$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = L$, $a=y$, and $b=(x - 2)$. So $L(x)=\sqrt{60^{2}+(x - 2)^{2}}=\sqrt{3600+(x - 2)^{2}}$.

Step2: Set $L(x)$ equal to 90

If $L(x)=90$, then $\sqrt{3600+(x - 2)^{2}}=90$. Square both sides of the equation: $3600+(x - 2)^{2}=90^{2}=8100$.

Step3: Solve for $(x - 2)^{2}$

Subtract 3600 from both sides: $(x - 2)^{2}=8100 - 3600=4500$.

Step4: Solve for $x-2$

Take the square root of both sides: $x - 2=\pm\sqrt{4500}=\pm30\sqrt{5}$. Since $x$ represents a distance, we take the positive value, so $x - 2 = 30\sqrt{5}\approx30\times2.236 = 67.08$.

Step5: Solve for $x$

Add 2 to both sides: $x=2 + 30\sqrt{5}\approx2+67.08=69.1$.

Answer:

(a) $L(x)=\sqrt{3600+(x - 2)^{2}}$
(b) $69.1$