Question

how far you can see on a clear day depends on where you are. on the ocean or on flat land the distance (d) to the horizon in miles can be found by using the formula d = √(1.5h). the variable h stands for the height of your eye above the land (in feet). use the formula to answer each question. measure or estimate the height of your own eye to the nearest tenth of a foot. how far out can you see from a beach? how far away would the horizon be if you were standing on top of an 80 - foot tower? (remember to add the height of your eye to the towers height.) how far is the horizon from the top of a 2000 - foot mountain? how far could you see from a plane flying three miles up? when the area of a circle is known, the diameter (d) can be found by using the formula d = 2√(a/3.14). a small - size pizza covers about 80 square inches. what is its diameter (to the nearest tenth of an inch)? what would be the diameter of a pizza twice as large (twice as much to eat)?

Explanation:

1. For the beach - seeing distance:

• Step1: Substitute height value
• Substitute \(h = 5.5\) into \(d=\sqrt{1.5h}\), getting \(d=\sqrt{1.5\times5.5}\).
• Step2: Calculate square - root
• Calculate \(\sqrt{8.25}\approx2.9\) miles.

1. For the 80 - foot tower:

• Step1: Calculate total height
• Find \(h=80 + 5.5 = 85.5\) feet.
• Step2: Substitute into formula
• Substitute \(h = 85.5\) into \(d=\sqrt{1.5h}\), getting \(d=\sqrt{1.5\times85.5}\).
• Step3: Calculate square - root
• Calculate \(\sqrt{128.25}\approx11.3\) miles.

1. For the 2000 - foot mountain:

• Step1: Substitute height value
• Substitute \(h = 2000\) into \(d=\sqrt{1.5h}\), getting \(d=\sqrt{1.5\times2000}\).
• Step2: Calculate square - root
• Calculate \(\sqrt{3000}\approx54.8\) miles.

1. For the plane flying three miles up:

• Step1: Convert miles to feet
• Since 1 mile = 5280 feet, \(h = 3\times5280=15840\) feet.
• Step2: Substitute into formula
• Substitute \(h = 15840\) into \(d=\sqrt{1.5h}\), getting \(d=\sqrt{1.5\times15840}\).
• Step3: Calculate square - root
• Calculate \(\sqrt{23760}\approx154.1\) miles.

1. For the small - size pizza:

• Step1: Substitute area value
• Substitute \(A = 80\) into \(d = 2\sqrt{\frac{A}{3.14}}\), getting \(d=2\sqrt{\frac{80}{3.14}}\).
• Step2: Calculate square - root and multiply
• First, calculate \(\sqrt{\frac{80}{3.14}}\approx5.047\), then \(d = 2\times5.047\approx10.1\) inches.

1. For the pizza twice as large:

• Step1: Calculate new area
• \(A_{new}=2\times80 = 160\) square inches.
• Step2: Substitute new area value
• Substitute \(A = 160\) into \(d = 2\sqrt{\frac{A}{3.14}}\), getting \(d=2\sqrt{\frac{160}{3.14}}\).
• Step3: Calculate square - root and multiply
• First, calculate \(\sqrt{\frac{160}{3.14}}\approx7.14\), then \(d = 2\times7.14\approx14.3\) inches.

Answer:

1. For the beach - seeing distance:

• Let's assume the height of the eye above the land \(h = 5.5\) feet. Using the formula \(d=\sqrt{1.5h}\), we have \(d=\sqrt{1.5\times5.5}=\sqrt{8.25}\approx 2.9\) miles.

1. For the 80 - foot tower:

• Assume the height of the eye is \(h_{eye}=5.5\) feet. The total height \(h = 80 + 5.5=85.5\) feet. Then \(d=\sqrt{1.5\times85.5}=\sqrt{128.25}\approx11.3\) miles.

1. For the 2000 - foot mountain:

• Using the formula \(d=\sqrt{1.5h}\), with \(h = 2000\) feet, \(d=\sqrt{1.5\times2000}=\sqrt{3000}\approx54.8\) miles.

1. For the plane flying three miles up:

• Since 1 mile = 5280 feet, three miles \(h=3\times5280 = 15840\) feet. Then \(d=\sqrt{1.5\times15840}=\sqrt{23760}\approx154.1\) miles.

1. For the small - size pizza:

• Given \(A = 80\) square inches and the formula \(d = 2\sqrt{\frac{A}{3.14}}\). Substitute \(A = 80\) into the formula: \(d=2\sqrt{\frac{80}{3.14}}=2\sqrt{25.4777}\approx2\times5.047\approx10.1\) inches.

1. For the pizza twice as large:

• If the new pizza has an area \(A_{new}=2\times80 = 160\) square inches. Using the formula \(d = 2\sqrt{\frac{A}{3.14}}\), substitute \(A = 160\) into it. \(d=2\sqrt{\frac{160}{3.14}}=2\sqrt{50.955}\approx2\times7.14\approx14.3\) inches.