a wall 10 feet high is 8 feet from a house. find the length of the shortest ladder that will reach the house when one end rest on the ground outside the wall.
Draw a diagram. Let x be the distance from the wall to the ladder's foot.
Let y be the length of the ladder.
Let θ be the angle the ladder makes with the ground.
tanθ = 10/x
secθ = y/(x+8)
tan^2θ + 1 = sec^2θ
100/x^2 + 1 = y^2/(x+8)^2
or,
y^2 = (x^2+100)(x+8)^2/x^2
y' = 2(x+8)(x^3-100)/x^3
y'=0 when x=-8 or 2∛100 = 9.28
y = 17.28/9.28 * sqrt(9.28^2 + 10^2) = 25.4 feet
Well, I'll have to climb a few steps to solve this one! Let's see... considering we have a wall 10 feet high and the ladder resting on the ground outside the wall, we essentially have a right-angled triangle. The height of the wall is the vertical leg, and the distance from the wall to the house is the horizontal leg.
To find the length of the ladder (which would be the hypotenuse of the triangle), we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, applying the theorem to our situation, we have:
(length of ladder)^2 = (height of wall)^2 + (distance to the house)^2
(length of ladder)^2 = 10^2 + 8^2
(length of ladder)^2 = 100 + 64
(length of ladder)^2 = 164
To find the length of the ladder, we need to take the square root:
length of ladder ≈ √164
Now, if you'll excuse me, I need to hire a clown mathematician to calculate that for me! *circus music plays*
(length of ladder) ≈ 12.81 feet
So, the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall is approximately 12.81 feet. Time to climb up and enjoy the view!
To find the length of the shortest ladder that will reach the house, we can use the Pythagorean theorem.
Let's consider the ladder as the hypotenuse of a right triangle, with one side being the height of the wall (10 feet) and the other side being the distance from the wall to the house (8 feet).
Using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can calculate the length of the ladder:
Length of the ladder^2 = Height of the wall^2 + Distance from the wall to the house^2
Length of the ladder^2 = 10^2 + 8^2
Length of the ladder^2 = 100 + 64
Length of the ladder^2 = 164
To find the length of the ladder, we need to take the square root of 164:
Length of the ladder = √164
Length of the ladder ≈ 12.81 feet
Therefore, the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall is approximately 12.81 feet.
To find the length of the shortest ladder, we can use the Pythagorean theorem. Let's assume that the ladder is the hypotenuse of a right triangle, with the wall being the vertical side and the distance between the wall and the house being the horizontal side.
Using the Pythagorean theorem, we have:
Ladder^2 = Wall^2 + Distance^2
Plugging in the given values:
Ladder^2 = 10^2 + 8^2
Ladder^2 = 164
To find the length of the ladder, we take the square root of both sides:
Ladder = √164
Ladder ≈ 12.81 feet
Therefore, the length of the shortest ladder that will reach the house when one end rests on the ground outside the wall is approximately 12.81 feet.