a 13- foot ladder is placed so that it reaches to a point on the wall that is 2 feet higher than twice the distance from the base of the wall to the base of the ladder. How high does the ladder reach?
x is the distance from base to wall
x^2 + (2x+2)^2 = 13^2
Now, you can expand that and solve for x, but the obvious hint is the 13' ladder. One of the common integer-sided triangles is the 5-12-13, where the lengths just exactly fit our problem.
The base is 5' from the wall, and the ladder top is 12' off the ground.
To determine how high the ladder reaches, we'll need to break down the problem into smaller steps.
Step 1: Define the variables:
Let's assume the height the ladder reaches is "H."
Let's also assume the distance from the base of the wall to the base of the ladder is "x."
Step 2: Set up the equation:
According to the problem, the ladder reaches a point on the wall that is 2 feet higher than twice the distance from the base of the wall to the base of the ladder.
This can be represented as:
H = 2x + 2
Step 3: Substitute relevant values:
We know the ladder is 13 feet long, so the total height it reaches is 13 feet. We can substitute this value for "H" in our equation:
13 = 2x + 2
Step 4: Solve for x:
To solve for "x," we'll need to isolate it on one side of the equation.
Subtract 2 from both sides of the equation:
13 - 2 = 2x
11 = 2x
Divide both sides of the equation by 2:
11/2 = x
Step 5: Calculate the height of the ladder:
Now that we have the value of "x," we can substitute it back into the equation to find the height of the ladder.
H = 2x + 2
H = 2(11/2) + 2
H = 11 + 2
H = 13
Therefore, the ladder reaches a height of 13 feet.