john's paint crew knows from experience that its 18 ft ladder is particularly stable when the distance from the ground to the top of the ladder is 5 ft more than the distance from the building to the base of the ladder. in this position, how far up the building does the ladder reach?

Functions?

This is a geometry problem.

The ladder up against the wall forms a right triangle.

x = base
x + 5 = opposite side
18 = hypotenuse

Using the Pythagorean theorem,
x^2 + (5 + x)^2 = 18^2

Solve for x

5 + x = how far up the blding the ladder reaches.

let the base be x ft

then the height is x+5 ft

solve x^2 + (x+5)^2 = 18^2

14.98?

x = 9.989983874

Use that answer to check your work.

(9.979983874)^2 + (9.979983874 + 5)^2 = 18^2

The height is exactly 14.979983874 ft.

To determine how far up the building the ladder reaches, we can set up an equation based on the given information.

Let's assume that the distance from the building to the base of the ladder is 'x' feet. Since we know that the distance from the ground to the top of the ladder is 5 feet more than the distance from the building to the base of the ladder, we can express it as 'x + 5' feet.

According to the problem, the length of the ladder is 18 feet. Now, we can use the Pythagorean theorem to relate the height, base, and ladder length.

According to the Pythagorean theorem, the square of the hypotenuse (the ladder in this case) is equal to the sum of the squares of the other two sides (the height and the base). Therefore:

length of ladder^2 = height^2 + base^2
18^2 = (x + 5)^2 + x^2

Now we can solve this equation to find the value of 'x', which represents the distance from the building to the base of the ladder.

324 = (x + 5)^2 + x^2

Expanding and simplifying the equation:

324 = x^2 + 10x + 25 + x^2

Combining like terms:

2x^2 + 10x + 25 - 324 = 0
2x^2 + 10x - 299 = 0

Now we can solve this quadratic equation. You can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 2, b = 10, and c = -299

Plugging the values into the quadratic formula:

x = (-10 ± √(10^2 - 4 * 2 * -299)) / (2 * 2)

Simplifying further:

x = (-10 ± √(100 + 2392)) / 4
x = (-10 ± √2492) / 4

Approximating the square root:

x = (-10 ± 49.92) / 4

Now we have two potential solutions:

x1 = (-10 + 49.92) / 4
x1 = 39.92 / 4
x1 ≈ 9.98 ft

x2 = (-10 - 49.92) / 4
x2 = -59.92 / 4
x2 ≈ -14.98 ft

Since the distance cannot be negative in this scenario, we discard the negative value. Therefore, the distance from the building to the base of the ladder is approximately 9.98 feet. Now, to find how far up the building the ladder reaches, we can substitute this value back into our initial equation:

Height = x + 5
Height ≈ 9.98 + 5
Height ≈ 14.98 feet

Thus, the ladder reaches approximately 14.98 feet up the building.