Question

jordan measures the distance around the edge of a circular dartboard as 56.52 inches. then she measures the distance from the center of the dartboard to its edge as 9 inches. jordan estimates the value of π by finding the ratio of 56.52 to 9. is this estimate correct? use the drop - down menus to explain. click the arrows to choose an answer from each menu. the value of π is equal to the ratio of the circumference of the dartboard to its jordans estimate involves the ratio of the circumference to its therefore, the estimate is choose...

Explanation:

Step1: Recall Circle Formulas

The circumference \( C \) of a circle is \( C = 2\pi r \) or \( C=\pi d \) (where \( d \) is diameter, \( d = 2r \)). The ratio for \( \pi \) is \( \pi=\frac{C}{d} \) (or \( \pi=\frac{C}{2r} \)). Here, the distance around the edge is circumference \( C = 56.52 \) inches, and the distance from center to edge is radius \( r = 9 \) inches, so diameter \( d=2\times9 = 18 \) inches.

Step2: Calculate \( \pi \) from Jordan's Estimate

Jordan uses ratio \( \frac{56.52}{9} \). Wait, but \( \pi=\frac{C}{d}=\frac{C}{2r} \). Let's compute \( \frac{56.52}{18} \) (since \( d = 18 \)): \( \frac{56.52}{18}=3.14 \). But Jordan used \( \frac{56.52}{9} \). Wait, maybe misstep: Wait, circumference \( C = 2\pi r \), so \( \pi=\frac{C}{2r} \). Given \( C = 56.52 \), \( r = 9 \), so \( 2r = 18 \). So \( \pi=\frac{56.52}{18}=3.14 \). But Jordan's ratio is \( \frac{56.52}{9}=6.28 \), which is \( 2\pi \). Wait, maybe the problem's drop - down menus: First, the value of \( \pi \) is equal to the ratio of circumference to diameter (since \( C=\pi d\Rightarrow\pi=\frac{C}{d} \)). Diameter is \( 2\times \text{radius}=18 \) inches. Jordan's estimate: she used ratio of circumference (56.52) to radius (9). Let's compute \( \frac{56.52}{9}=6.28 \), and \( 2\pi = 6.28 \) (since \( \pi\approx3.14 \), \( 2\pi\approx6.28 \)). But \( \pi \) is \( \frac{C}{d}=\frac{56.52}{18}=3.14 \). Wait, maybe the problem's structure: The first drop - down: "The value of \( \pi \) is equal to the ratio of the circumference of the dartboard to its \(\boldsymbol{\text{diameter}}\)". Second: "Jordan's estimate involves the ratio of the circumference to its \(\boldsymbol{\text{radius}}\)". Then, calculate \( \frac{56.52}{9}=6.28 \), and \( \pi\approx3.14 \), so \( 6.28 = 2\pi \), so her ratio is \( 2\pi \), not \( \pi \). So the estimate is incorrect because she used radius instead of diameter. Let's verify: If we use \( \pi=\frac{C}{2r} \), \( \frac{56.52}{2\times9}=\frac{56.52}{18}=3.14 \), which is \( \pi \). Jordan used \( \frac{56.52}{9}=6.28 \), which is \( 2\pi \), so her ratio is for \( 2\pi \), not \( \pi \). So the correct ratio for \( \pi \) is circumference to diameter. Jordan used circumference to radius. Then, the estimate: since \( \pi\approx3.14 \), and her ratio is 6.28 (which is \( 2\pi \)), so her estimate (using ratio of C to r) is not equal to \( \pi \), so the estimate is incorrect.

Answer:

(Assuming drop - down options: First blank: "diameter", Second blank: "radius", Third blank: "incorrect" (since \( \frac{56.52}{9}=6.28 = 2\pi
eq\pi \))). The key is: \( \pi=\frac{\text{Circumference}}{\text{Diameter}} \), Jordan used \( \frac{\text{Circumference}}{\text{Radius}} \), so her estimate (ratio \( \frac{56.52}{9}=6.28 \)) is not equal to \( \pi \) (which is \( 3.14 \)), so the estimate is incorrect.