A 22m ladder and a 20m ladder were leaned against a building.
The bottom of the longer ladder was 4m farther from the building than the bottom of the shorter ladder, but both ladders reached the same distance up the building.
Find this distance to the nearest tenth.
Select one:
a. 15.3m
b. 19.2m
c. 18.1m
d. 12.2m
Draw a picture. If the bottom of the shorter ladder is x from the building, and both ladders reach to height h, then
x^2+h^2 = 20^2
(x+4)^2 + h^2 = 22^2
so, eliminating h,
20^2 - x^2 = 22^2 - (x+4)^2
Now solve for x, and then you can get h
To solve this problem, we can use the principles of right triangles and Pythagorean theorem. Let's assign variables to the unknown distances.
Let x be the distance from the bottom of the shorter ladder to the building.
And x + 4 will be the distance from the bottom of the longer ladder to the building.
Now, let's use the Pythagorean theorem to set up an equation:
(x + 4)^2 + 20^2 = x^2 + 22^2
Simplifying the equation:
(x^2 + 8x + 16) + 400 = x^2 + 484
Combine like terms:
x^2 + 8x + 416 = x^2 + 484
Subtracting x^2 from both sides:
8x + 416 = 484
Subtracting 416 from both sides:
8x = 68
Dividing both sides by 8:
x = 8.5
Therefore, the distance from the bottom of the shorter ladder to the building is 8.5m.
And the distance from the bottom of the longer ladder to the building is (8.5 + 4) = 12.5m.
Since both ladders reach the same distance up the building, the answer is 12.5m to the nearest tenth.
So, the correct option is d. 12.2m.