A contractor leans a 13-foot ladder against a building. The distance from the ground to the top of the ladder is 7 feet more than the distance from the building to the base of the ladder. Find the distance from the building to the base of the ladder and the distance from the ground to the top of the ladder.
To solve this problem, we can set up a system of equations based on the given information. Let's define two variables:
Let x represent the distance from the building to the base of the ladder.
Let y represent the distance from the ground to the top of the ladder.
According to the problem, we know that the ladder is 13 feet long. Therefore, using the Pythagorean theorem, we can write the equation:
x^2 + y^2 = 13^2
Additionally, the problem states that "the distance from the ground to the top of the ladder is 7 feet more than the distance from the building to the base of the ladder." This can be written as:
y = x + 7
Now we have a system of two equations:
x^2 + y^2 = 13^2
y = x + 7
To find the values of x and y, we can substitute the second equation into the first equation.
Substituting y in terms of x, we get:
x^2 + (x + 7)^2 = 13^2
Expanding and simplifying the equation:
x^2 + (x^2 + 14x + 49) = 169
x^2 + x^2 + 14x + 49 = 169
2x^2 + 14x + 49 - 169 = 0
2x^2 + 14x - 120 = 0
Divide the equation by 2 to simplify it:
x^2 + 7x - 60 = 0
Now we can solve this quadratic equation for x. We can either factor the equation or use the quadratic formula:
Factoring the equation:
(x - 5)(x + 12) = 0
Setting each factor equal to zero, we have:
x - 5 = 0 or x + 12 = 0
Solving each equation:
x = 5 or x = -12
Since the distance cannot be negative, we discard x = -12 as a solution. Therefore, x = 5.
Now that we have the value for x, we can substitute it back into the second equation to find y:
y = x + 7
y = 5 + 7
y = 12
Therefore, the distance from the building to the base of the ladder is 5 feet, and the distance from the ground to the top of the ladder is 12 feet.
base distance --- x
height where the ladder touches --- x+7
x^2 + (x+7)^2 = 13^2
2x^2 + 14x + 49 = 169
x^2 + 7x - 60 = 0
(x+12)(x-5) = 0
x = -12 , which makes no sense
or
x = 5
finish it up