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Math Help - Calculus Word Problems

  1. #1
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    Calculus Word Problems

    1) The number of bus riders from the suburbs to downtown per day is represented by 1200(1.50-x), where x is the fare in dollars. What fare will maximize the total revenue?

    #2 A large social part y follow the mathematical progression
    N(t)=30t-t^2, where t is the time in minutes since the party began and N is the number of separate conversation occurring. At what time in a party do the most conversations occur? what is the maximum number of interactions?

    #3A new cottage is built across the river and 300 m downstream from the nearest telephone station. The river is 120 m wide. In order to wire the cottage for phone service, wire will be laid across the river underwater, and along the edge of the river above ground. The cost to laywire under water is $15 per m and the cost to lay wire above ground is 10 per m. How much wire should be laid under water to minimize the cost?
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  2. #2
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    Quote Originally Posted by notoriousmc View Post
    1) The number of bus riders from the suburbs to downtown per day is represented by 1200(1.50-x), where x is the fare in dollars. What fare will maximize the total revenue?
    The total revenue will be the product of the per-person rate (the fare) and the number of riders.

    What stands for "the per-person rate"?

    What stands for "the number of riders"?

    What is their product?

    Then differentiate, and maximize. (You will end up with a negative quadratic function, which you learned about back in algebra. From what you learned about graphing back then, you know that there is exactly one max/min point, being the vertex, and that, for a negative quadratic, the vertex is the maximum.)

    Quote Originally Posted by notoriousmc View Post
    #2 A large social party follows the mathematical progression N(t)=30t-t^2, where t is the time in minutes since the party began and N is the number of separate conversation occurring. At what time in a party do the most conversations occur? what is the maximum number of interactions?
    For what does the t-value stand?

    For what does the N-value stand?

    Differentiate, set equal to zero, and maximize. (Since the function is an upside-down quadratic, you know, from back in algebra, that the only critical point will of course be the vertex, and will be the maximum.)

    At the maximum, what does "t" represent?

    At the maximum, what does "N" represent?

    Quote Originally Posted by notoriousmc View Post
    #3 A new cottage is built across the river and 300 m downstream from the nearest telephone station. The river is 120 m wide. In order to wire the cottage for phone service, wire will be laid across the river underwater, and along the edge of the river above ground. The cost to laywire under water is $15 per m and the cost to lay wire above ground is 10 per m. How much wire should be laid under water to minimize the cost?
    Draw two horizonal lines, representing the sides of the river. Draw a vertical line between, representing the width. Label the point on the lower line as A and the upper line as B; label AB as having a length of 120.

    On the upper line, to the right of B, label a point as C; label the distance between B and C as 300.

    The "nearest telephone station" is at A; the cottage is at C.

    Draw a slanty line from A to some point D between B and C. This is the optimal path that you need to find. Label BD as x.

    What then is (the expression for) the value of DC?

    ABD is a right triangle with legs having lengths 120 and x. What then is (the expression for) the length of AD?

    You are given that the cost, along the slanty line AD, is $15 per meter. What then is (the expression for) the cost of AD?

    You are given that the cost, along DC, is $10 per meters. What then is (the expression for) the cost of DC?

    What then is the expression for the total cost?

    Find the value of x that minimizes this cost.

    If you get stuck, please reply showing how far you have gotten in working through the steps. Thank you!
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