a fence 8 ft tall stands on level ground and runs parallel to a tall building. if the fence is 1 ft from the building, find the length of the shortest ladder that will extend from the ground over the fence to the wall of the building. (hint #1: if L represents the length of the ladder, the quantity L is minimized when the quantity L2 is minimized, so you do not have to concern yourself with thte square root-just minimize L2 and that will minimize L. Hint#2: use similar triangles)
L/ |
/|8|
/_|_|
..|1|
Make a diagram
let the foot of the ladder be x ft from the fence
let the ladder reach y ft above the ground
I see similar triangle so set up a ratio
8/x = y/(x+1)
xy = 8x+8
y = (8x+8)/x
let the length of the ladder be L
L^2 = (x+1)^2 + y^2
= (x+1)^2 + [(8x+8)/x]^2
= x^2+2x+1 + 64 + 128/x + 64/x^2
2L dL/dx = 2x + 2 - 128/x^2 - 128/x^3
= 0 for a min of L
2x + 2 - 128/x^2 - 128/x^3 = 0
multiply by x^3
2x^4 + 2x^3 - 128x - 128 = 0
2x^3(x+1) - 128(x+1) = 0
(x+1)(2x^3 - 128) = 0
x = -1 , not likely
or
2x^3=128
x^3=64
x=4
sub into L^2
L^2 = 5^2 + 10^2 = 125
I minimized L^2
Make a diagram
let the foot of the ladder be x ft from the fence
let the ladder reach y ft above the ground
I see similar triangle so set up a ratio
8/x = y/(x+1)
xy = 8x+8
y = (8x+8)/x
let the length of the ladder be L
L^2 = (x+1)^2 + y^2
= (x+1)^2 + [(8x+8)/x]^2
= x^2+2x+1 + 64 + 128/x + 64/x^2
2L dL/dx = 2x + 2 - 128/x^2 - 128/x^3
= 0 for a min of L
2x + 2 - 128/x^2 - 128/x^3 = 0
multiply by x^3
2x^4 + 2x^3 - 128x - 128 = 0
2x^3(x+1) - 128(x+1) = 0
(x+1)(2x^3 - 128) = 0
x = -1 , not likely
or
2x^3=128
x^3=64
x=4
sub into L^2
L^2 = 5^2 + 10^2 = 125
I minimized L^2
To find the length of the shortest ladder, we can apply the concept of similar triangles. Let's break down the given scenario:
1. First, draw a diagram to visualize the situation. You have a fence (represented by a vertical line) that is 8 ft tall, standing 1 ft away from a tall building (represented by another vertical line).
2. Since we are looking for the length of the shortest ladder, let's assume the ladder starts from the ground, extends over the fence, and reaches the wall of the building (represented by a diagonal line).
3. Now, let's consider the similar triangles formed by the ladder, the fence, and the building. There are two similar triangles we can work with:
- The larger triangle formed by the ladder, the fence, and the ground.
- The smaller triangle formed by the ladder, the wall of the building, and the ground.
4. Let's label the lengths in the diagram:
- The length of the ladder (which we need to find) is L.
- The length of the fence is 8 ft.
- The distance between the fence and the building is 1 ft.
5. Since the two triangles are similar, we can set up a proportion between their corresponding sides:
- For the larger triangle: L/8 = (L + 1)/8
- For the smaller triangle: L/x = (L + 1)/8
6. Solve the proportion for the larger triangle:
- Cross-multiply to get L = (L + 1)
- Simplify to get L = 1
7. Substitute the value of L = 1 in the proportion for the smaller triangle:
- 1/x = (1 + 1)/8
- Simplify to get 1/x = 2/8
- Cross-multiply to get 8 = 2x
- Simplify to get x = 4 ft
8. The value of x represents the length of the ladder required to extend from the ground over the fence to the wall of the building. Therefore, the length of the shortest ladder is 4 ft.
By applying the concept of similar triangles and setting up the appropriate proportion, we were able to find the desired length of the ladder.