A tool shed 250cm high and 100 cm deep is built against a wall. Calculate the shortest ladder that can reach from the ground, over the shed, to the wall behind.
How can I solve this?
To solve this problem, we can use the concept of the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we can consider the ladder as the hypotenuse, with the height of the shed being one side, and the depth of the shed being the other side. Let's label these as follows:
Height of shed (height): 250 cm
Depth of shed (depth): 100 cm
Length of ladder (ladder): unknown
Using the Pythagorean theorem, we can write the equation:
height^2 + depth^2 = ladder^2
Substituting the given values, we have:
250^2 + 100^2 = ladder^2
Simplifying:
62500 + 10000 = ladder^2
72500 = ladder^2
To find the length of the ladder, we need to take the square root of both sides:
√72500 = √ladder^2
269.26 = ladder
Therefore, the shortest ladder that can reach from the ground, over the shed, to the wall behind is approximately 269.26 cm in length.
To solve this problem, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle formed by the ladder, the height of the shed, and the distance from the base of the shed to the wall.
Let's calculate this step by step:
Step 1:
The ladder forms the hypotenuse of the right triangle.
Step 2:
The height of the shed is 250 cm, which will be one of the legs of the triangle.
Step 3:
The distance from the base of the shed to the wall will be the other leg of the triangle.
Now, using the Pythagorean theorem, we can set up the equation:
ladder^2 = height^2 + distance^2
Substituting the given values:
ladder^2 = 250^2 + 100^2
ladder^2 = 62500 + 10000
ladder^2 = 72500
Taking the square root of both sides to find the length of the ladder:
ladder = √72500
ladder ≈ 269.85 cm (rounded to two decimal places)
Therefore, the shortest ladder that can reach from the ground, over the shed, to the wall behind is approximately 269.85 cm.
make a sketch, let the shed touch the wall at A and the ground at B
Let the ladder touch the wall at P and the ground at Q
Let R be the point where the ladder touches the shed
let O be the bottom of the wall meeting the ground.
I am also going to scale down the values given by a factor of 50 to keep the numbers smaller.
let's use trig: let angle Q = angle P = Ø
then cosØ = 2/PR ----> PR = 2secØ
and sinØ = 5/RQ ---> RQ = 5cscØ
PQ = PR+RQ
= 2secØ + 5cscØ
d(PQ)/dØ = 2secØtanØ - 5cscØcotØ
= 0 for a min of PQ
2secØtanØ = 5cscØcotØ
2(1/cosØ)(sinØ/cosØ) = 5(1/sinØ)(cosØ/sinØ)
2sinØ/cos^2 Ø = 5cosØ/sin^2 Ø
sin^3 Ø/cos^3 Ø = 5/2
tan^3 Ø = 2.5
tanØ = 1.3572...
Ø = 53.627°
PQ = ......
let me know what you got, don't forget to multiply PQ by 50, my scaling factor.