Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes?

Please see the attached file for the full problems.
1. Show by integration that the first two energy eigenstates for the Harmonic Oscillator are orthogonal functions.
2. For eignstate ψ_2 (x) of the harmonic oscillator, show that...

Please see the attached file for the diagram associated with this problem.
Question: An airplane is flying at an altitude of 8 miles towards a point directly over an observer. If the speed of the plane is 615 miles per hour, find the rate at which the angle of observation, theta, changing by at the moment when the angle is 2

I need help solving the following problems. Can you explain to me how to solve these problems?
Find the derivative of y = log(2 + sin〖x)〗
Find the derivative of f(x) = sinh2(3x)
Find the derivative of y = x^(x-1)
Find the derivative of y = ln((1+e^x)/(1-e^x ))
Evaluate ∫_0^(π/4)▒tan

This solution shows how to solve for various calculus problems, including differentiation of functions using the product rule, the quotient rule, and the chain rule, as well as how to calculate integrals.

I need to find the center of mass (gravity) for problems two and tree. all integration procedure be shown .
The answers required are. (the x and the y)
2a) x bar =
2b) y bar =
3a) x bar =
3b) y bar =

One cannot be involved in U.S. health care for very long these days without hearing the term "integration." Specifically, we hear a great deal about "horizontal integration" and "vertical integration" of health care services. So, what do these terms really mean? Do you feel that such integration represents positive change for he

10 pts. 2. A farmer with 750 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible area of the four pens?
10 pts. 3. Use Newton's method to find the approximations from x1 to x3 to find the third approximation of t

Please show how to solve each of the following problems.
Find the antiderivative (integral)
6. (x^2/the square root of [x^3-4])dx
8. (x^2 - 2)^3 2x dx
9. sin^3(x)dx
10. x^3/x^2 + 1 dx
11. 1/xln x dx
12. ln x/x dx
14. 2x + 1/square root of [x + 4] dx