Question

draw a diagram and solve the following problems using the pythagorean theorem.

1. youre locked out of your house and the only open window is on the second floor, 25 feet above the ground. you need to borrow a ladder from one of your neighbors. theres a bush along the edge of the house, so youll have to place the ladder 10 feet from the house. what length of ladder do you need to reach the window?
2. a television screen measures approximately 15 in. high and 19 in. wide. a television is advertised by giving the approximate length of the diagonal of its screen. how should this television be advertised?
3. jackson is 54 miles west of lazy r resort. ontario is 31 miles south of jackson. a land developer proposes building a shortcut road to directly connect ontario and lazy r.
4. you need to construct a ramp to roll a cart from your garage into the back of your truck. the truck is 8 feet from the garage. the back of the truck is 3 feet above the ground. how long does the ramp have to be (in feet)?

Explanation:

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Problem 19

Step1: Define Pythagorean variables

Let vertical side $a=25$ ft, horizontal side $b=10$ ft, ladder (hypotenuse) $c$.
Pythagorean theorem: $c^2=a^2+b^2$

Step2: Substitute values and calculate

$c^2=25^2+10^2=625+100=725$

Step3: Solve for hypotenuse

$c=\sqrt{725}=\sqrt{25\times29}=5\sqrt{29}\approx26.93$ ft
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Problem 20

Step1: Define Pythagorean variables

Let height $a=15$ in, width $b=19$ in, diagonal (hypotenuse) $c$.
Pythagorean theorem: $c^2=a^2+b^2$

Step2: Substitute values and calculate

$c^2=15^2+19^2=225+361=586$

Step3: Solve for hypotenuse

$c=\sqrt{586}\approx24.21$ in
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Problem 21

Step1: Define Pythagorean variables

Let west distance $a=54$ mi, south distance $b=31$ mi, shortcut (hypotenuse) $c$.
Pythagorean theorem: $c^2=a^2+b^2$

Step2: Substitute values and calculate

$c^2=54^2+31^2=2916+961=3877$

Step3: Solve for hypotenuse

$c=\sqrt{3877}\approx62.27$ mi
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Problem 22

Step1: Define Pythagorean variables

Let vertical height $a=3$ ft, horizontal distance $b=8$ ft, ramp (hypotenuse) $c$.
Pythagorean theorem: $c^2=a^2+b^2$

Step2: Substitute values and calculate

$c^2=3^2+8^2=9+64=73$

Step3: Solve for hypotenuse

$c=\sqrt{73}\approx8.54$ ft
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Answer:

1. Ladder length: $\boldsymbol{5\sqrt{29}\approx26.93}$ feet
2. Television advertised as: $\boldsymbol{\approx24.21}$ inch diagonal
3. Shortcut road length: $\boldsymbol{\sqrt{3877}\approx62.27}$ miles
4. Ramp length: $\boldsymbol{\sqrt{73}\approx8.54}$ feet