A 17- foot ladder is leaning against a wall, and begins to slide. The top of the ladder is falling at a rate of 2 feet per second at the instant that the bottom of the ladder is 8 feet from the wall. What is the rate of change of the distance between the bottom of the ladder and the bottom of the wall at this instant?
Also what is the rate of change of the area of the triangle made by the ladder?
let h be the height along the wall, f the distance along the floor.
17^2=h^2+f^2
0=2h dh/dt + 2f df/dt
find df/dt, all else is given.note that dh/dt is negative, it is goind down.
area? Area=1/2 b*h=1/2 f*h
dA/dt=1/2 h df/dt + 1/2 f dh/dt
again, dh/dt is negative.
To find the rate of change of the distance between the bottom of the ladder and the bottom of the wall, we can use the Pythagorean theorem.
Let's denote the distance between the bottom of the ladder and the bottom of the wall as x, and the distance between the top of the ladder and the bottom of the wall as y. We are given that y is changing at a rate of 2 feet per second.
According to the Pythagorean theorem, we have the relationship:
x^2 + y^2 = 17^2
Differentiating both sides of this equation with respect to time (t) using implicit differentiation, we get:
2x(dx/dt) + 2y(dy/dt) = 0
At the instant when the bottom of the ladder is 8 feet from the wall, we have:
x = 8 and y = 17
Substituting these values into the equation, we can solve for dx/dt, which represents the rate of change of x:
2(8)(dx/dt) + 2(17)(2) = 0
16(dx/dt) + 34 = 0
16(dx/dt) = -34
dx/dt = -34/16
dx/dt = -17/8
Therefore, the rate of change of the distance between the bottom of the ladder and the bottom of the wall at this instant is -17/8 feet per second.
To find the rate of change of the area of the triangle made by the ladder, we can use the formula for the area of a triangle:
Area = 0.5 * base * height
At any given time, the base of the triangle is fixed at 8 feet, since the bottom of the ladder remains 8 feet from the wall. Therefore, the only changing variable is the height of the triangle, which is the y-coordinate.
Differentiating the area formula with respect to time using implicit differentiation, we get:
d(Area)/dt = 0.5 * base * (dh/dt)
Substituting the values, we have:
base = 8
dh/dt = 2 (given)
Therefore, the rate of change of the area of the triangle made by the ladder is:
d(Area)/dt = 0.5 * 8 * 2
d(Area)/dt = 8 square feet per second.
To find the rate of change of the distance between the bottom of the ladder and the bottom of the wall, we can use the Pythagorean theorem, which relates the lengths of the sides of a right triangle.
Let's denote the length from the bottom of the ladder to the bottom of the wall as x, and the length from the bottom of the ladder to the top of the ladder as y. According to the problem, at the instant when the bottom of the ladder is 8 feet from the wall, we know that y = 17 - x (since the ladder is 17 feet long).
We are given that dy/dt (the rate of change of y, which is the top of the ladder falling) is 2 ft/s. We need to find dx/dt (the rate of change of x), which is what the question is asking for.
Using the Pythagorean theorem, we have:
x^2 + y^2 = (17 - x)^2
x^2 + (17 - x)^2 = 289 - 34x + x^2
Simplifying, we get:
2x^2 - 34x + 256 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 2, b = -34, and c = 256. Plugging in these values, we find:
x = (-(-34) ± √((-34)^2 - 4(2)(256))) / (2(2))
x = (34 ± √(1156 - 2048)) / 4
x = (34 ± √(-892)) / 4
Since the square root of a negative number is undefined in the context of this problem, we can conclude that there is no real solution for x. Therefore, at the instant when the bottom of the ladder is 8 feet from the wall, the rate of change of the distance between the bottom of the ladder and the bottom of the wall is undefined.
Moving on to the second part of the question, we need to find the rate of change of the area of the triangle made by the ladder. The area of a triangle is given by the formula A = (1/2)bh, where b is the base (x) and h is the height (y). Substituting y = 17 - x, we have:
A = (1/2)x(17 - x)
A = (17/2)x - (1/2)x^2
To find the rate of change of the area, we can differentiate A with respect to time (t):
dA/dt = (17/2)(dx/dt) - (1/2)[(2x)(dx/dt)]
Since we already know that dx/dt is undefined (as established earlier), the rate of change of the area of the triangle made by the ladder is also undefined at this instant.