"A 9-foot wall stands 30 ft from the building. find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall".
So I believe I need to start with the Pythagorean Theorem formula, but where do I need to go from there? All help is appreciated!
I would use trig.
Make a sketch, showing the ladder touching the top of the fence.
I see 2 similar right-angled triangles, both with a base angle of θ.
Let the hypotenuse of the triangle touching the wall be L1
Let the hypotenuse of the triangle touching the ground be L2
for L1, cosθ = 30/L1 ----> L1 = 30/cosθ
for L2, sinθ = 9/L2 ----> L2 = 9/sinθ
Let L = L1 + L2
= 30/cosθ + 9/sinθ = 30(cosθ)^-1 + 9(sinθ)^-1
dL/dθ = -30(cosθ)^-2 (-sinθ) - 9(sinθ)^-2 (cosθ)
= 30sinθ/cos^2 θ - 9cosθ/sin2 θ
= 0 for a min of L
30sinθ/cos^2 θ = 9cosθ/sin2 θ
30 sin^3 θ = 9cos^3 θ
sin^3 θ/cos^3 θ = 3/10 = .3
tan^3 θ = .3
tanθ = (.3)^(1/3) = .66943...
θ = 33.7996... °
then L1 = 30/cos 33.799.. ° = 36.1 ft
and L2 = 9/sin 33.799.. ° = 16.2 ft
L = 36.1 + 16.2 ft = 52.3 ft
Phewww!!!!
check my calculations please
Or, if you want to use the Pythagorean Theorem,
L = length of beam
x = distance from wall to the foot of the beam
h = height where beam touches the wall
Then, using similar triangles,
h/(x+30) = 9/x
h = 9(x+30)/x
L^2 = (x+30)^2 + h^2 = (x+30)^2 + 81((x+30)/x)^2
dL/dx = 0 when x = ∛2470 ≈ 13.44
h = 9(13.44+30)/13.44 = 29.08
Thus, L^2 = 43.44^2 + 29.08^2
L = 52.27
which agrees with Reiny's result >whew<
Thank you so much, Mr. Reiny, you are amazing!
I checked the calculations and they were correct. 52.3 ft was one of the answer choices!
Well, let's see if I can provide some "beaming" assistance!
Given that you have a 9-foot tall wall and the beam needs to reach from the ground outside the wall to the side of the building, we can form a right triangle.
The height of the wall is one leg of the triangle, and the distance from the wall to the building is the hypotenuse. The length of the beam will be the other leg of the triangle.
Using the Pythagorean Theorem:
(a^2) + (b^2) = (c^2)
where a and b are the legs of the triangle and c is the hypotenuse, we can solve for the length of the beam.
In this case, the height of the wall (a) is 9 ft and the distance from the wall to the building (c) is 30 ft. So, we have:
(9^2) + (b^2) = (30^2)
81 + (b^2) = 900
Now, let's solve for b:
b^2 = 900 - 81
b^2 = 819
b ≈ √819
b ≈ 28.63 ft
Therefore, the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall is approximately 28.63 feet.
And remember, don't be too "beamish" when telling others about this solution!
To solve this problem, you are on the right track by starting with the Pythagorean Theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this situation, we can assume the wall, ground outside the wall, and the beam form a right-angled triangle. Let's label the sides of the triangle:
- Let x represent the length of the beam, which is the hypotenuse.
- Let y represent the height of the wall.
- Let z represent the distance between the wall and the building.
Based on the given information, we have:
y = 9 ft (height of the wall)
z = 30 ft (distance between the wall and the building)
Now, we can apply the Pythagorean Theorem using the formula:
x^2 = y^2 + z^2
Substituting the values we know:
x^2 = (9 ft)^2 + (30 ft)^2
Simplifying:
x^2 = 81 ft^2 + 900 ft^2
x^2 = 981 ft^2
To find x, we take the square root of both sides:
x = √(981 ft^2)
x ≈ 31.35 ft
Therefore, the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall is approximately 31.35 feet.