Question

period date

1. a car travels 6 miles due north then makes a turn due west. it travels 7 miles west. how far is the car from its starting - point?
2. michelle delivers books to school libraries. her truck has a slide - out ramp for unloading the books. the top of the ramp is 3 feet above the ground. the ramp itself is 6 feet long. what is the horizontal distance the ramp reaches?
3. pria has a 15 - foot ladder. the safety instructions recommend he should have the base of the ladder 6 feet from the base of the wall he will lean the ladder against. how high will the ladder reach on the wall?
4. a local businessman bought a square plot of land. the sides of the lot measure 33 feet on each side. he decides to split the lot into two equal - sized right triangles by putting a fence down the diagonal. approximately how many feet of fencing will he need?
5. a rectangular prism is 5 inches long, 8 inches wide and 10 inches tall. what is the length of its longest diagonal?
6. chris is mailing his friend a poster that has been rolled up in a long tube. he has a box that measures 20 inches by 8 inches by 4 inches. what is the maximum length the rolled poster can be?

Explanation:

Step1: Identify right - triangle problems

All these problems can be solved using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse of a right - triangle and \(a\) and \(b\) are the other two sides.

Step2: Problem 1

The car's path forms a right - triangle with legs \(a = 6\) miles and \(b=7\) miles. Using the Pythagorean theorem \(d=\sqrt{6^{2}+7^{2}}=\sqrt{36 + 49}=\sqrt{85}\approx9.2\) miles.

Step3: Problem 2

The ramp forms a right - triangle with hypotenuse \(c = 6\) feet and one leg \(a = 3\) feet. We want to find the other leg \(b\). Using the Pythagorean theorem \(b=\sqrt{6^{2}-3^{2}}=\sqrt{36 - 9}=\sqrt{27}\approx5.2\) feet.

Step4: Problem 3

The ladder, the wall and the ground form a right - triangle with hypotenuse \(c = 15\) feet and one leg \(a = 6\) feet. We want to find the height \(b\) on the wall. Using the Pythagorean theorem \(b=\sqrt{15^{2}-6^{2}}=\sqrt{225 - 36}=\sqrt{189}\approx13.7\) feet.

Step5: Problem 4

For a square of side \(s = 33\) feet, the diagonal \(d\) of the square (which is the hypotenuse of a right - triangle formed by two sides of the square) is given by \(d=\sqrt{33^{2}+33^{2}}=\sqrt{2\times33^{2}}=33\sqrt{2}\approx46.7\) feet.

Step6: Problem 5

For a rectangular prism with length \(l = 5\) inches, width \(w = 8\) inches and height \(h = 10\) inches, the length of the longest diagonal \(D\) is given by \(D=\sqrt{5^{2}+8^{2}+10^{2}}=\sqrt{25 + 64+100}=\sqrt{189}\approx13.7\) inches.

Step7: Problem 6

For a box with dimensions \(l = 20\) inches, \(w = 8\) inches and \(h = 4\) inches, the length of the longest diagonal \(D\) (which is the maximum length of the rolled poster) is given by \(D=\sqrt{20^{2}+8^{2}+4^{2}}=\sqrt{400 + 64 + 16}=\sqrt{480}\approx21.9\) inches.

Answer:

1. Approximately 9.2 miles
2. Approximately 5.2 feet
3. Approximately 13.7 feet
4. Approximately 46.7 feet
5. Approximately 13.7 inches
6. Approximately 21.9 inches