9.) The top of a 25 foot ladder is sliding down a vertical wall at a constant rate of 4 feet per minute. When the top of the ladder is 15 feet off the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

Question

Bytelearn · Accepted Answer

Triangle Definition: We are dealing with a right triangle where the ladder represents the hypotenuse, the wall represents one leg, and the distance from the wall to the bottom of the ladder represents the other leg. We can use the Pythagorean theorem to relate these sides. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a^2 + b^2 = c^2.Pythagorean Theorem Application: Given that the ladder is 25 feet long (hypotenuse) and the top of the ladder is 15 feet off the ground (one leg), we can find the initial distance (x) between the bottom of the ladder and the wall using the Pythagorean theorem: x^2 + 15^2 = 25^2.Calculate Initial Distance: Let's calculate the initial distance (x) between the bottom of the ladder and the wall: x^2 + 225 = 625.Isolate x^2: Subtract 225 from both sides to isolate x^2: x^2 = 625 - 225, x^2 = 400.Find x Value: Take the square root of both sides to solve for x: x = √400, x = 20 feet. So, the initial distance from the wall to the bottom of the ladder is 20 feet.Related Rates Approach: Now, we need to find the rate of change of the distance between the bottom of the ladder and the wall. As the ladder slides down, the top of the ladder is moving at a rate of 4 feet per minute. We can use related rates to find the rate at which x is changing. Let's denote the rate of change of the distance from the wall as dx/dt.Differentiate Pythagorean Theorem: Differentiating both sides of the Pythagorean theorem with respect to time (t), we get: 2x(dx/dt) + 2(15)(-4) = 0. The negative sign for the rate of change of the height of the ladder (-4 feet per minute) indicates that the height is decreasing.Solve for dx/dt: Now we can solve for dx/dt: 2x(dx/dt) = 2(15)(4), 2x(dx/dt) = 120.Substitute x Value: We already found that x is 20 feet, so we can substitute that value into the equation: 2(20)(dx/dt) = 120.Final Rate of Change: Solve for dx/dt: 40(dx/dt) = 120, (dx/dt) = 120 / 40, (dx/dt) = 3 feet per minute. This is the rate of change of the distance between the bottom of the ladder and the wall.

The top of a $25$ foot ladder is sliding down a vertical wall at a constant rate of $4$ feet per minute. When the top of the ladder is $15$ feet off the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

Full solution

More problems from Rate of travel: word problems

Related topics

The top of a 252525 foot ladder is sliding down a vertical wall at a constant rate of 444 feet per minute. When the top of the ladder is 151515 feet off the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

The top of a $25$ foot ladder is sliding down a vertical wall at a constant rate of $4$ feet per minute. When the top of the ladder is $15$ feet off the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?