A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 9 ft less than the length of the ladder.how high up the side of the building is the top of the ladder if that distance is 2 ft less than the length of the ladder? the ladder reaches ___ ft up the wall.
Isn't this an application of the Pythagorean theorem?
(ladder)^2=(height^2 + (base)^2
but ladder=height+2
and base=ladder-9 or base=height+7
(h+2)^2=(h^2)+(h+7)^2
solve for h.
To find the height the ladder reaches up the wall, let's set up an equation.
Let's assume the length of the ladder is 'L' feet.
According to the problem, the distance from the bottom of the ladder to the building is 9 feet less than the length of the ladder, which gives us L - 9.
We are also given that the distance from the top of the ladder to the wall is 2 feet less than the length of the ladder, which gives us L - 2.
Now, we can create a right-angled triangle with the ladder as the hypotenuse, the distance to the wall as one leg, and the distance to the building as the other leg.
According to Pythagoras' theorem, the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.
So, we have the equation:
(L - 9)^2 + (L - 2)^2 = L^2
Simplifying the equation:
L^2 - 18L + 81 + L^2 - 4L + 4 = L^2
Combining like terms:
2L^2 - 22L + 85 = L^2
Subtracting L^2 from both sides:
L^2 - 22L + 85 = 0
This is a quadratic equation that can be factored or solved using the quadratic formula. After solving the equation, we find that the length of the ladder is approximately 20 feet.
Now, we can substitute the value of L back into the equation for the height the ladder reaches up the wall:
L - 2 = 20 - 2 = 18 feet
Therefore, the ladder reaches 18 feet up the side of the building.