a ladder which is 15ft leans against a wall. suppose that when the bottom of the ladder is x feet away from the wall,the bottom is being pushed towards the wall at a rate of 1/2x feet per second. how fast is the top of the ladder rising at the moment the bottom is 5 feet from the wall?
so i did pythagerous and i figured out the other side which was 14.14, since that is the y side or the height of the wall at a point where the ladder is 5 feet from the wall
from there i ddid the derivative of
x^2 + y^ 2 = c^2
which was
2x(dx/dt) + 2y(dy/dt) = 0 ( this is 0 cuz the hypothenouse is constant)
and then i subbed in the numbers and solved for dy/dt
the numbers i subbed in were
a = 5
y = 14.14
dx/dt= 0.5
and im getting -.0.176 as my answer but the correct answer for some reason is 0.884 , can someoen tell me what im doing wrong?
thanks a lot
x/dt = - 1/2 it is getting smaller.
5 (-.5) + 14.14 (dy/dt) = 0
dy/dt = .176
so I agree with you except for the sign.
okay so other than the sign are you saying what im doing is correct? and the answer is most likely wrong?
Yes.
To find the rate at which the top of the ladder is rising, we need to differentiate the equation x^2 + y^2 = c^2, where x represents the distance of the bottom of the ladder from the wall, y represents the height reached by the ladder on the wall, and c represents the length of the ladder.
Taking the derivative with respect to time, we get:
2x(dx/dt) + 2y(dy/dt) = 0
Now, let's substitute the given values:
x = 5 ft
y = 14.14 ft (calculated using the Pythagorean theorem with x = 5 ft)
(dx/dt) = 1/2x = 1/2(5) = 0.5 ft/s (given)
When we plug in these values into the derivative equation, we get:
2(5)(0.5) + 2(14.14)(dy/dt) = 0
Simplifying this equation gives:
5 + 28.28(dy/dt) = 0
Rearranging and solving for dy/dt, we have:
28.28(dy/dt) = -5
(dy/dt) = -5/28.28
Evaluating this expression gives approximately -0.176 ft/s.
Based on your calculations, it seems that you made an arithmetic mistake. However, it's worth noting that speed and velocity are signed quantities, so the answer of -0.176 ft/s is correct. The negative sign indicates that the top of the ladder is lowering, while a positive answer would mean it is rising. Therefore, the correct answer is -0.176 ft/s, not 0.884 ft/s.