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?

Explanation:

Step1: Identify the problem type for question 12

This is a right - triangle problem using the Pythagorean theorem. Let the distance from the wall be $a = 5$ feet, the length of the ladder be $c=13$ feet, and the height on the wall be $b$.
According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, so $b=\sqrt{c^{2}-a^{2}}$.

Step2: Calculate the height on the wall

Substitute $a = 5$ and $c = 13$ into the formula:
$b=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$ feet.

Step3: Identify the problem type for question 13

For a rectangle with length $l = 75$ meters and width $w = 40$ meters, to find the length of the diagonal $d$, we use the Pythagorean theorem. In a rectangle, if the sides are $l$ and $w$, and the diagonal is $d$, then $d^{2}=l^{2}+w^{2}$.

Step4: Calculate the diagonal of the rectangle

Substitute $l = 75$ and $w = 40$ into the formula:
$d=\sqrt{75^{2}+40^{2}}=\sqrt{5625+1600}=\sqrt{7225}=85$ meters.

Answer:

1. 12
2. 85