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Math  /  Geometry

QuestionGraph units are in meters.
What is the distance to the hole for the player who is farthest from the hole? Round the final answer to the nearest tenth of a meter. Do not round intermediate calculations.

Studdy Solution

STEP 1

1. The graph is in a Cartesian coordinate system with units in meters.
2. The positions of Eric, Brooke, Cam, and the hole are given by their coordinates.
3. The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) can be calculated using the distance formula.

STEP 2

1. Recall the distance formula.
2. Calculate the distance from Eric to the hole.
3. Calculate the distance from Brooke to the hole.
4. Calculate the distance from Cam to the hole.
5. Determine the maximum distance.
6. Round the final answer to the nearest tenth.

STEP 3

Recall the distance formula:
d=(x2−x1)2+(y2−y1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

STEP 4

Calculate the distance from Eric (−6,5)(-6, 5) to the hole (4,2)(4, 2):
dEric=(4−(−6))2+(2−5)2 d_{\text{Eric}} = \sqrt{(4 - (-6))^2 + (2 - 5)^2} =(4+6)2+(−3)2 = \sqrt{(4 + 6)^2 + (-3)^2} =102+9 = \sqrt{10^2 + 9} =100+9 = \sqrt{100 + 9} =109 = \sqrt{109}

STEP 5

Calculate the distance from Brooke (10,9)(10, 9) to the hole (4,2)(4, 2):
dBrooke=(4−10)2+(2−9)2 d_{\text{Brooke}} = \sqrt{(4 - 10)^2 + (2 - 9)^2} =(−6)2+(−7)2 = \sqrt{(-6)^2 + (-7)^2} =36+49 = \sqrt{36 + 49} =85 = \sqrt{85}

STEP 6

Calculate the distance from Cam (8,−6)(8, -6) to the hole (4,2)(4, 2):
dCam=(4−8)2+(2−(−6))2 d_{\text{Cam}} = \sqrt{(4 - 8)^2 + (2 - (-6))^2} =(−4)2+(8)2 = \sqrt{(-4)^2 + (8)^2} =16+64 = \sqrt{16 + 64} =80 = \sqrt{80}

STEP 7

Determine the maximum distance:
Compare 109\sqrt{109}, 85\sqrt{85}, and 80\sqrt{80}.
109≈10.4403 \sqrt{109} \approx 10.4403 85≈9.2195 \sqrt{85} \approx 9.2195 80≈8.9443 \sqrt{80} \approx 8.9443
The maximum distance is 109\sqrt{109}.

STEP 8

Round the maximum distance to the nearest tenth:
109≈10.4 \sqrt{109} \approx 10.4
The distance to the hole for the player who is farthest from the hole is:
10.4 meters \boxed{10.4 \text{ meters}}

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