A reservoir is shaped like a vertical circular cone with the tip down (like an ice cream cone). Its height is 25m, the radius of its upper circular face is 8m, and it is filled with water up to 17m. The purpose of this question is to determine the total work required to pump all the water to a height of 4m above the top of the tank. We recall that the density of water is = 1000 kg/m^3 and that the gravitational acceleration is g = 9.8 m/s^2.

(a) Let x be the height (in meters) measured from the tip to the bottom of the tank. Find an expression for the approximate volume V (x) of a thin layer of water between x and x + x m. To get the maximum score: You must clearly draw and label a diagram. You must show all your work and briefly explain your answer.

(b) What is the approximate work required to pump the thin layer of water (described in part (a)) 4 m above the top of the tank? Briefly justify your answer.

(c) Give a definite integral that calculates the total work required to pump all the water of the reservoir 4 m above its top. Don't evaluate the integral - just write it down.

Each of the following functions admits a MacLaurin series expansion. For each series:

i. Give its representation in Maclaurin series using summation notation.

ii. Write explicitly its first four (4) nonzero terms. Write the exact numbers for the coefficients. Your answers must be well justified. Show all your steps!

(a) f(x) = x^10 sin(2x)

(b) g(x) = (2 x)^5

Please can we do it It's two diffrent exercise do the first please it's a urgence