Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground. She only has a 20 ft. ladder. How far from the base of the house should she set up the ladder so that the top of the ladder is exactly at the bottom of the window?(1 point)
ft.
To solve this problem, we can use the Pythagorean Theorem. The ladder, the distance from the base of the house to the ladder, and the distance from the top of the ladder to the window form a right triangle.
Let x be the distance from the base of the house to the ladder.
According to the Pythagorean Theorem, the sum of the squares of the two shorter sides is equal to the square of the longest side (the hypotenuse).
Using this, we can set up the equation:
x^2 + 16^2 = 20^2
Simplifying this equation:
x^2 + 256 = 400
Subtracting 256 from both sides:
x^2 = 144
Taking the square root of both sides:
x = ±12
Since the problem states that the ladder should be set up in front of the house, we can ignore the negative solution.
Therefore, the ladder should be set up 12 ft. from the base of the house.
Use the image to answer the question.
A rectangle is formed from two right triangles. The base of the rectangle measures 80 yards and is labeled Johnson Avenue. The top of the rectangle is labeled Edison Avenue. The left side joining both the avenues is labeled 41st street and measures 60 yards. The right side of the rectangle is labeled 42nd street. A bank is located at the vertex of 41st street and Edison Avenue. The hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street is labeled park.
Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90° angles, how much shorter is it for Sylvia to walk through the park?
(1 point)
If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be
yards shorter.
To find out how much shorter it is for Sylvia to walk through the park instead of straight down Johnson Avenue and 41st Street, we need to find the length of the hypotenuse (the park route) and compare it to the sum of the lengths of Johnson Avenue and 41st Street.
Using the Pythagorean Theorem, we can find the length of the hypotenuse:
(c^2) = (a^2) + (b^2)
In this case, a = 80 yards and b = 60 yards.
(c^2) = (80^2) + (60^2)
(c^2) = 6400 + 3600
(c^2) = 10000
Taking the square root of both sides to find c:
c = √10000
c = 100 yards
Therefore, the length of the hypotenuse (the park route) is 100 yards.
The sum of the lengths of Johnson Avenue and 41st Street is 80 + 60 = 140 yards.
To find out how much shorter it is for Sylvia to walk through the park, we subtract the length of the park route from the sum of the lengths of Johnson Avenue and 41st Street:
140 - 100 = 40
Therefore, it is 40 yards shorter for Sylvia to walk through the park instead of straight down Johnson Avenue and 41st Street.
To solve this problem, we can use the concept of similar triangles.
Let's define a few variables to help us with the calculations:
- The distance from the base of the ladder to the house is "x" feet.
- The height of the window from the ground is given as 16 feet.
Now, we can set up a proportion between the two similar triangles formed by the ladder and the window:
(height of ladder) / (distance from base of ladder to house) = (height of window) / (distance from base of ladder to bottom of window)
By substituting the values we know into this proportion, we get the following equation:
20 / x = 16 / x + 20
To solve this equation, we can cross-multiply:
20(x + 20) = 16x
20x + 400 = 16x
4x = 400
Dividing both sides of the equation by 4, we find:
x = 100
Therefore, Camila should set up the ladder 100 feet from the base of the house to reach the bottom of the window.