A ladder is resting against a wall. The top of the ladder touches the wall at a height of 15 feet and the length of the ladder is one foot more than twice the distance from the wall. Find the distance from the wall to the bottom of the ladder. (Hint: Use the Pythagorean Theorem.)
And please can you show me all your work.
make a sketch.
let the distance of the foot from the wall be x ft
label the base x
the length of the ladder (the hypotenuse) is
one foot more than twice the distance from the wall
---- 2x + 1
mark the hypotenuse 2x + 1
and of course the height is 15
so in your right-angled triangle
15^2 + x^2= (2x+1)^2
225 + x^2 = 4x^2 + 4x + 1
0 = 3x^2 + 4x - 224
Using the quadratic formula:
x = (-4 ± √2704)/6
= (-4 ± 52)/6
= 8 or a negative, which we would reject
So the ladder is 8 ft from the wall
check:
height = 15
base = 8
ladder = 17
is 15^2 + 8^2 = 17^2 ?? , YES!
To find the distance from the wall to the bottom of the ladder, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's break down the given information:
- The top of the ladder touches the wall at a height of 15 feet.
- The length of the ladder is one foot more than twice the distance from the wall.
Let's assume the distance from the wall to the bottom of the ladder is "x" feet.
According to the problem, the length of the ladder is one foot more than twice the distance from the wall. So the length of the ladder can be expressed as (2x + 1) feet.
We now have a right-angled triangle where one side is the height of the ladder touching the wall (15 feet), another side is the distance from the wall to the bottom of the ladder (x feet), and the hypotenuse is the length of the ladder [(2x + 1) feet].
Applying the Pythagorean Theorem, we can write the equation as:
(15^2) + (x^2) = ((2x + 1)^2)
Now we can solve this equation to find the value of x, which represents the distance from the wall to the bottom of the ladder.