Question

1. the bottom of a ladder must be placed 6 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:

12.

Step1: Apply Pythagorean theorem

Let the height above the ground be $h$. The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, where $c = 13$ (length of the ladder) and $a = 5$ (distance from the wall). So $5^{2}+h^{2}=13^{2}$.

Step2: Simplify the equation

$25 + h^{2}=169$. Then $h^{2}=169 - 25=144$.

Step3: Solve for $h$

Taking the square - root of both sides, $h=\sqrt{144}=12$ feet.

Step1: Apply Pythagorean theorem for rectangle

For a rectangle with length $l = 78$ meters and width $w = 40$ meters, the diagonal $d$ is found using the Pythagorean theorem $d^{2}=l^{2}+w^{2}$.

Step2: Calculate $l^{2}+w^{2}$

$l^{2}=78^{2}=6084$ and $w^{2}=40^{2}=1600$. Then $l^{2}+w^{2}=6084 + 1600=7684$.

Step3: Find the diagonal

$d=\sqrt{7684}=87.66$ meters $\approx88$ meters.

Step1: Apply Pythagorean theorem

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. According to the Pythagorean theorem $x^{2}+12^{2}=15^{2}$.

Step2: Simplify the equation

$x^{2}+144 = 225$. Then $x^{2}=225 - 144 = 81$.

Step3: Solve for $x$

Taking the square - root of both sides, $x=\sqrt{81}=9$ feet.

Answer:

12

13.