A 25 foot ladder is leaning against a house when its base starts to slide away by the time the base is 7 feet from the house, so we're going to let X equal our seven. The base is moving away at the rate of 24 feet per second. So our DX DT is going to be 24 feet per second. So now we can figure out what our Y value is going to be at that specific time. So y ^2 + 7 ^2 = 25 ^2 or Y is equaling sqrt 25 ^2 - 7 ^2 or Y equal 24. Now we have a relationship of the sides of X ^2 + y ^2 equal 25 ^2. Because 25 is a constant, the latter is not going to change the length. So when we take the derivative of that, we get 2XD XDT +2 YDYDT equaling 0 because the derivative of a constant 0. Solving for DYDT, we're going to subtract the 2XD XDT, divide each side by two, and then divide by Y. So we get negative X / y DXDT. We were given that X is 7, so -7 we found that Y was 24, and we were given that DXDT was 24. So we know that the height is changing. It's decreasing, or it's getting smaller by 7 feet per second. The next part of this says, how's the area changing? Well, the area of a triangle is 1/2 base height. So we're going to take the derivative of that. So the derivative of a = 1/2. We have to do the product rule. So we're going to take the derivative of B times the H plus 1/2 B times the derivative of the H. So DADT equal 1/2 DB DT times H + 1/2 BDHDT. So DA DT1 half the DB DT. The base is really just our X, so our DXDT was 24. Our height is really just our Y, so that's also 24 plus 1/2. The base is the X or seven, and the DHDT is really our change in our Y. So it's -7 So when we compute that, we get 527 halves feet squared per second. The last part of this problem asks for the angle is changing. So we know that tangent Theta is y / X. So the derivative in terms of time of tangent as secant squared Theta and the derivative in terms of time of y / X. We have to use the quotient rule. So the derivative of the top DYDT times the bottom X minus the derivative of the bottom DXDT times the top Y all over the bottom squared. If we take that secant squared to the other side, that's really just going to be cosine squared, and looking at our triangle, we know that cosine is adjacent over hypotenuse. So 7 / 25 ^2 times the DYDT was -7 the X was 7 minus the DXDT was 24, the Y was 24 all over the X ^2 or 7 ^2. When we compute that, we get -1 feet per second. Oh, we get -1 radians per second, not feet, because we're doing an angle.