Question

a graph of circumference vs. diameter yields a numerical slope of π. what does the slope tell us about the relationship between ac and ad? a string is wrapped around the earth, forming a circle at the equator of circumference 25,000 miles. if 15 ft are added to the string, how far will the string now stand above the earth?

Explanation:

Step1: Recall the formula for the circumference of a circle

The formula for the circumference of a circle is $C = 2\pi r=\pi d$, where $C$ is the circumference, $r$ is the radius and $d$ is the diameter. When we graph $C$ vs $d$, the equation of the line is $y = mx + b$, where $y = C$, $x = d$, $m=\pi$ and $b = 0$. The slope $\pi$ means that for every unit increase in the diameter $d$, the circumference $C$ increases by $\pi$ units.

Step2: Set up equations for the original and new - length strings

Let the original radius of the Earth (at the equator) be $r_1$ and the new radius be $r_2$. The original circumference $C_1=2\pi r_1 = 25000$ miles. The new circumference $C_2=C_1 + 15$ feet. First, convert 25000 miles to feet. Since 1 mile = 5280 feet, $C_1=25000\times5280$ feet. Then $C_2=25000\times5280+15$ feet.

Step3: Use the circumference formula to find the radii

Since $C_1 = 2\pi r_1$ and $C_2 = 2\pi r_2$, we have $r_1=\frac{C_1}{2\pi}$ and $r_2=\frac{C_2}{2\pi}$. The distance $h$ that the string stands above the Earth is $h=r_2 - r_1$. Substitute $r_1$ and $r_2$ into the formula: $h=\frac{C_2}{2\pi}-\frac{C_1}{2\pi}=\frac{C_2 - C_1}{2\pi}$. Since $C_2 - C_1=15$ feet, then $h=\frac{15}{2\pi}\approx\frac{15}{2\times3.14}\approx2.39$ feet.

Answer:

The string will stand approximately 2.39 feet above the Earth.