A fence 8 feet tall stands on level ground and runs parallel to a tall building. If the fence is 1 foot from the building, find the shortest ladder that will extend from the ground over the fence to the wall of the building.
This solution is correct but isn't the last step wrong because isn't it (8+32)^2/4^2 instead of (8+32)^4/4?
I assume you made a sketch.
let the distance from the fence to the foot of the ladder be x
let the height of the ladder up the wall be h
by ratios:
8/x = h/(1+x)
hx = 8(1+x
h = (8+8x)/x
let the length of the ladder be L
L^2 = h^2 + (1+x)^2
= (8+8x)^2/x^2 + (1+x)^2
= (64 + 128x + 64x^2)/x^2 + 1 + 2x + x^2
= 64/x^2 + 128/x + 64 + 1 + 2x + x^2
2L dL/dx = -128/x^3 - 128/x^2 + 2 + 2x
= 0 for a min of L
times -x^3
128 + 128x - 2x^3 - 2x^4 = 0
128(1+x) - 2x^3(1+x) = 0
(1+x)(2x^3 - 128) = 0
x = -1 , which makes no sense
or
2x^3 - 128 = 0
x^3 = 64
x = 4
So L^2 = (8 + 32)^2 /4 + (1+4)^2
= 400 + 25
L^2 = 425
L = √425 = 5√17 = appr 20.6 ft
check my arithmetic.
To find the length of the shortest ladder that will extend from the ground over the fence to the wall, we can use the Pythagorean theorem.
Let's call the distance from the ground to the top of the building "h" and the distance from the bottom of the ladder (where it touches the ground) to the wall of the building "d".
The ladder, the height of the fence, and the distance from the fence to the building form a right-angled triangle.
Using the Pythagorean theorem, we have the equation:
h^2 = d^2 + 8^2
Since we know the distance between the fence and the building is 1 foot, we have d = 1 foot.
Substituting this value into the equation, we have:
h^2 = 1^2 + 8^2
Simplifying further, we have:
h^2 = 1 + 64
h^2 = 65
Taking the square root of both sides, we have:
h = sqrt(65)
Therefore, the shortest ladder that will extend from the ground over the fence to the wall of the building is approximately sqrt(65) feet long.
To find the shortest ladder that will extend from the ground over the fence to the wall of a building, we can use the Pythagorean theorem.
Let's call the height of the ladder "h" and the distance from the base of the ladder to the building "d".
According to the problem, the fence is 8 feet tall, and it is 1 foot away from the building. Therefore, the height of the ladder h will be the sum of the height of the fence (8 feet) and the height it needs to reach the top of the building. Thus, h = 8 feet + height of the building.
The distance d will be the same as the distance between the fence and the building (1 foot).
Now, using the Pythagorean theorem, we have:
h^2 = d^2 + (8 feet)^2
Plugging in the values we know:
h^2 = (1 foot)^2 + (8 feet)^2
Simplifying the equation:
h^2 = 1 square foot + 64 square feet
h^2 = 65 square feet
Taking the square root of both sides:
h = √65 feet
Therefore, the shortest ladder that will extend from the ground over the fence to the wall of the building is approximately √65 feet.