Towns A and B are located near a straight river as shown in Figure 1 (not...
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Towns A and B are located near a straight river as shown in Figure 1 (not drawn to scale). A pumping station is to be built on the river's edge with pipes extending straight to the two towns. Your task is to find where the pumping station should be built (i.e. what is the value of x) to minimize the total length of the pipe required to connect both towns to the pumping station. 15 km 7 km x km Pumping station Figure 1: Proposed pumping station. B 4 km (a) Find a function, f(x), for the the total length of the pipe required as a function of x. (8 marks) (b) Using calculus and the expression found in (a) to determine, algebraically, the value of a (in kilometres) that gives the minimum total length of the pipes needed. What is the minimum total length of the pipes required? Check your value of a by substitution into the derivative. Also verify that you have found the minimum by using an appropriate calculus test. (42 marks) Towns A and B are located near a straight river as shown in Figure 1 (not drawn to scale). A pumping station is to be built on the river's edge with pipes extending straight to the two towns. Your task is to find where the pumping station should be built (i.e. what is the value of x) to minimize the total length of the pipe required to connect both towns to the pumping station. 15 km 7 km x km Pumping station Figure 1: Proposed pumping station. B 4 km (a) Find a function, f(x), for the the total length of the pipe required as a function of x. (8 marks) (b) Using calculus and the expression found in (a) to determine, algebraically, the value of a (in kilometres) that gives the minimum total length of the pipes needed. What is the minimum total length of the pipes required? Check your value of a by substitution into the derivative. Also verify that you have found the minimum by using an appropriate calculus test. (42 marks)
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