A contractor leans a 23-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. How far up the building is the ladder to the nearest tenth of a foot?
23^2=(x)^2 + (x+7)^2 x is the base distance.
solve for x.
so the base is 4.1? then i subtract that from 23?
math is important
Well, it sounds like this contractor has a "height" situation on their hands!
Let's call the distance from the building to the base of the ladder "x". According to the problem, the distance from the ground to the top of the ladder is 7 feet more than x.
So, the equation we have is: x + 7 + x = 23
To solve for x, let's simplify the equation: 2x + 7 = 23
Now, let's get rid of that pesky 7 by subtracting it from both sides: 2x = 16
Dividing both sides by 2, we find that x = 8
Therefore, the distance up the building to the nearest tenth of a foot is 8 + 7 = 15 feet.
So, it looks like the ladder reaches up 15 feet. That's a pretty "elevating" solution, don't you think?
To find the distance up the building that the ladder reaches, we'll need to set up a right triangle with the ladder as the hypotenuse and the ground-to-building distance and the height of the ladder as the other two sides.
Let's define the following variables:
- x: Distance from the building to the base of the ladder
- x + 7: Distance from the ground to the top of the ladder
- 23: Length of the ladder
According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (the longest side, which is the ladder in this case) is equal to the sum of the squares of the other two sides.
So, we can set up the equation:
x^2 + (x + 7)^2 = 23^2
Simplifying the equation, we get:
x^2 + x^2 + 14x + 49 = 529
Combining like terms, we have:
2x^2 + 14x + 49 = 529
Rearranging the equation, we have:
2x^2 + 14x - 480 = 0
To solve this quadratic equation, we can use the quadratic formula, which states that:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 2, b = 14, and c = -480.
Plugging in these values into the quadratic formula, we get:
x = (-14 ± √(14^2 - 4(2)(-480))) / (2(2))
Simplifying further:
x = (-14 ± √(196 + 3840)) / 4
x = (-14 ± √(4036)) / 4
x ≈ (-14 ± 63.56) / 4
x ≈ (49.56 / 4) or (-77.56 / 4)
x ≈ 12.39 or -19.39
Since distance cannot be negative, we discard the negative solution.
Therefore, x ≈ 12.39 feet.
To find the height up the building that the ladder reaches, we add 7 to x:
Height ≈ 12.39 + 7
Height ≈ 19.39 feet
Therefore, the ladder reaches approximately 19.39 feet up the building.