Question

a sporting goods company produces an archery target. the scale drawing of the plan shows the center at (-1.5, 3.5) and one point on the circle at (8, 9). what is the approximate area of the target in units on the scale drawing? use 3.14 for π, and round the answer to the nearest whole number. 3,219 units² 656 units² 1,513 units² 378 units²

Explanation:

Step1: Find the radius (distance between center and point)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \((x_1,y_1)=(-1.5,3.5)\) and \((x_2,y_2)=(8,9)\).
So, \(x_2 - x_1=8 - (-1.5)=9.5\) and \(y_2 - y_1=9 - 3.5 = 5.5\).
Then, \(r=\sqrt{(9.5)^2+(5.5)^2}=\sqrt{90.25 + 30.25}=\sqrt{120.5}\approx10.98\).

Step2: Calculate the area of the circle

The formula for the area of a circle is \(A=\pi r^2\). We use \(\pi = 3.14\) and \(r\approx10.98\).
So, \(A = 3.14\times(10.98)^2\approx3.14\times120.56\approx378.56\approx379\)? Wait, no, wait, maybe I miscalculated the radius. Wait, let's recalculate the radius:

Wait, \(x_2 - x_1=8 - (-1.5)=9.5\), \(y_2 - y_1=9 - 3.5 = 5.5\). Then \((9.5)^2=90.25\), \((5.5)^2 = 30.25\), sum is \(90.25+30.25 = 120.5\), square root of 120.5 is approximately 10.98? Wait, no, 10^2=100, 11^2=121, so \(\sqrt{120.5}\approx10.98\), which is approximately 11. Then area is \(3.14\times11^2=3.14\times121 = 379.94\approx380\)? But the options have 378. Wait, maybe my approximation of the radius is off. Wait, maybe I made a mistake. Wait, let's check the options. Wait, maybe the radius calculation:

Wait, \(8 - (-1.5)=9.5\), \(9 - 3.5 = 5.5\). Then \(9.5^2=90.25\), \(5.5^2=30.25\), sum is 120.5. \(\sqrt{120.5}\approx10.977\), so \(r\approx10.98\). Then area is \(3.14\times(10.98)^2\approx3.14\times120.56\approx378.56\), which rounds to 378. Ah, right, because 10.98 squared is approximately 120.5, 3.14*120.5=378.37, which rounds to 378.

Wait, so the correct answer is 378 units². Wait, but let's check again:

Distance between (-1.5,3.5) and (8,9):

\(\Delta x=8 - (-1.5)=9.5\)

\(\Delta y=9 - 3.5 = 5.5\)

\(r=\sqrt{(9.5)^2+(5.5)^2}=\sqrt{90.25 + 30.25}=\sqrt{120.5}\approx10.977\)

Area \(A = \pi r^2=3.14\times(10.977)^2\approx3.14\times120.5\approx378.37\approx378\). So the answer is 378 units².

Wait, but the options have 378 as one of them. So the correct option is the last one: 378 units².

Answer:

378 units² (corresponding to the option with 378 units²)