Question

You are buying a ladder for your 30 -foot-tall house. For safety, you would always like to ensure that the base of the ladder be placed at least 8 feet from the base of the house. What is the shortest ladder you can buy in order to be able to reach the top of your house?

Short answer

Answer: The shortest ladder needed is 32 feet long.

Step-by-step solution

Identify the right triangle

In this situation, we have a right triangle with one leg representing the height of the house (30 feet) and another leg representing the minimum distance between the base of the ladder and the base of the house (8 feet). The hypotenuse of this triangle will be the length of the ladder, which we need to find.

Apply the Pythagorean theorem

Using the Pythagorean theorem for right triangles, we can express the relationship between the legs and the hypotenuse as follows: a^2 + b^2 = c^2 where 'a' and 'b' are the legs of the triangle, and 'c' is the hypotenuse. In our case, 'a' is the height of the house (30 feet), 'b' is the minimum distance between the base of the ladder and the base of the house (8 feet), and 'c' is the length of the ladder that we need to find.

Solve for the ladder's length 'c'

Using the values for 'a' and 'b' given in the problem, we can plug them into the Pythagorean theorem equation and solve for 'c': (30)^2 + (8)^2 = c^2 900 + 64 = c^2 964 = c^2 Now we can find the value of 'c' by taking the square root of 964: c = sqrt(964)

Calculate and round up the length of the ladder

Now we need to calculate the square root of 964: c ≈ 31.048 Since we need an integer length for the ladder and it must be able to reach the top of the house, we should round up to the nearest whole number: c ≈ 32 Therefore, the shortest ladder needed to reach the top of the 30-foot-tall house while placing its base at least 8 feet away from the house is 32 feet long.