Question

1. the bottom of a ladder must be placed 5 feet from a wall. the ladder is 13 feet long. how far above the ground does the ladder touch the wall? 13. a soccer field is a rectangle 40 meters wide and 75 meters long. the coach asks players to run from one corner to the other corner diagonally across. what is that distance? 14. how far from the base of the house do you need to place a 15 - foot ladder so that it exactly reaches the top of a 12 - foot tall wall?

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with sides \(a\), \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Solve problem 12

Let the height on the wall be \(h\), the distance from the wall be \(a = 5\) feet and the length of the ladder be \(c=13\) feet. Using the Pythagorean theorem \(h=\sqrt{c^{2}-a^{2}}\). Substitute \(a = 5\) and \(c = 13\) into the formula: \(h=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\) feet.

Step3: Solve problem 13

The length of the soccer field is \(a = 75\) meters and the width is \(b = 40\) meters. The diagonal \(d\) of the rectangle is given by the Pythagorean theorem \(d=\sqrt{a^{2}+b^{2}}\). Substitute \(a = 75\) and \(b = 40\) into the formula: \(d=\sqrt{75^{2}+40^{2}}=\sqrt{5625+1600}=\sqrt{7225}=85\) meters.

Step4: Solve problem 14

Let the distance from the base of the house be \(x\), the height of the wall is \(a = 12\) feet and the length of the ladder is \(c = 15\) feet. Using the Pythagorean theorem \(x=\sqrt{c^{2}-a^{2}}\). Substitute \(a = 12\) and \(c = 15\) into the formula: \(x=\sqrt{15^{2}-12^{2}}=\sqrt{225 - 144}=\sqrt{81}=9\) feet.

Answer:

1. 12 feet
2. 85 meters
3. 9 feet