Calculus homework

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1. Let w(t) be the number of widgets produced at time t. Let the rate of production (widgets per week) for widgets at a company is given by

w'(t)= 50,000(1- (1000/(t+10)^2)     .

Integrate both sides of the previous differential equation from 0 to 10 to get the total production of widgets in the first 10 weeks of production. Assume that at week 0 no widgets were produced. Hint: you will need to apply the fundamental theorem of calculus and use a simple substitution. Show all details and steps. 

 

2. A small warehouse has extra space wants to rent storage cages made out of fencing. To make the cages the warehouse manager wants to use 500ft of fencing material to enclose a large rectangle and then divide it into 20 cages with fencing parallel to one side of the rectangle. Since the cages will be rented at $10 per ft2 use calculus to find the dimensions of the an individual cage that maximizes area and the maximum total income that all the cages will produce.

 

 

3. A trough is 10ft long and its ends have the shape of isoceles triangles that are 3ft across at the top and have a height of 1ft. Suppose the trough is being filled with water at a rate of 12ft3/min.

(a) Let, V (t) be the volume in trough in ft3 at time t minutes. Give a formula for V (t) in terms of h(t) where h(t) is height of the water in the trough in ft at time t minutes.

(b) Use your formula for V (t) and h(t) to find how fast the water level is rising when height of the water is .75ft. (that is, relate the rate of change of volume to the rate of change of height) 

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    PROF. NORMAX
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