Calculating Driveway Gravel Area and Optimizing Cardboard Box Volume

For each part of this problem, make sure to: 

  • introduce the question being asked, in your own words, 
  • explain your approach and the reasoning behind it,
  • show your computations on separate lines to the text,
  • conclude by explaining the result in the context of the problem
The driveway of a house is straight on three sides, with a decorative curce on the fourth side shown in the graph attached below. The axes are measured in meters. 

A) The owner of the house wants to fill the drivewat with gravel, but needs to know the area to be covered. use a left or right Riemann sum with six subintervals to give an over-estimete of the area. (You can use the nearest half-interger value for the height of the graph at any given point). 

B) The homeowner is an avid surfer, and reconizes the decorative curve as a sinusoidal wave, determing the graph given by the function f(x) = 4 + 3sin(x). Find the antiderivative of f(x) and use the fundemental theorem of calculus to find the actual area of the driveway. 

C) To help move the gravel, the homeowner intends using a think sheet of cardboard measuring 6 feet by 6 feet to make an open-topped box, by cutting out equal-sized squares of lengh x from the corners and folding up the four edges. Give an equation for the volume of the box as a function of the square lenghs x. 

D) Find the value of x that gives the maximum volume for the open-topped box. 
The answer is accepted.
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