Applications
1. Suppose that the number of people in your school that hear a rumor about you x hours after it is started is given by the function .
(a) What is the average rate at which the rumor is spreading between and ?
(b) What is the instantaneous rate at which the rumor is spreading after 4 hours?
(c) Is the rate at which the rumor is spreading increasing or decreasing? Hint: consider the graph of the function.
2. If gasoline costs $.80/L and you drive 24000 km per year then your annual cost for gasoline will be given by the function where x is the number of km/L that your car can travel.
(a) Find your annual gasoline costs if your car gets 3 km/l.
(b) Find your annual gasoline costs if your car gets 8 km/l.
(c) What is the average rate of change in your annual gasoline cost as x changes from 3 to 8.
(d) What is the instantaneous rate of change in your annual fuel costs when ?
3. Two numbers have a sum of 4. How must they be chosen in order to maximize their product? What is that maximum product?
4. What number exceeds its square by the greatest amount? What is that amount?
5. Sadie has 60 m of fencing that she plans to use to enclose a rectangular garden plot. One side of the garden will be against the barn, so she does not need to put a fence on that side. Find the dimensions of the plot that will maximize the area. What is the maximum area?
6. What are the dimensions of the largest rectangle that can be inscribed in a semi-circle of radius 60 m?
7. A rectangular box-shaped garbage can with a square base and an open top is to be constructed using exactly 2700 of material. Find the dimensions of the box that will provide the greatest possible volume.
8. An oil well has been discovered offshore at W, 200 m from S, the nearest point on the shoreline. Town T is located 1000 m along the shore from point S. A pipeline must be installed underwater from W to V and then along the shoreline from V to T. If it costs $500/m to run the pipe underwater and $200/m to run the pipe along the shore, how far from S should V be located to minimize the total cost of the pipeline?
9. Determine the velocity and acceleration functions for the given position functions.
(a)
(b)
(c)
10. A particle moves along the x-axis so that its position in metres after t seconds is given by the function . Find:
(a) The velocity and acceleration at any time t.
(b) The velocity when .
(c) The acceleration when .
(d) The position of the particle when the velocity is 66 m/s.
(e) The velocity of the particle when the acceleration is .
11. The length of a rectangle is increasing at a rate of 6 cm/s. If the area of the rectangle is not changing, at what rate is the width of the rectangle decreasing when the length is 14 cm and the width is 7 cm?
12. Water is being poured into a conical vase at a rate of . The diameter of the cone is 30 cm and the height of the cone is 25 cm. At what rate is the water level rising when the water’s depth is 20 cm?
13. A cylindrical tank has a radius of 3m and a depth of 10m. It is being filled at the rate of 5 m3/min. How fast is the surface rising?
Answer Key
1. Suppose that the number of people in your school that hear a rumor about you x hours after it is started is given by the function .
(a) What is the average rate at which the rumor is spreading between and ?
(b) What is the instantaneous rate at which the rumour is spreading after 4 hours?
(c) Increasing
2. If gasoline costs $.80/L and you drive 24000 km per year then your annual cost for gasoline will be given by the function where x is the number of km/L that your car can travel.
(a) Find your annual gasoline costs if your car gets 3 km/L.
(b) Find your annual gasoline costs if your car gets 8 km/L.
(c) What is the average rate of change in your annual gasoline cost as x changes from 3 to 8.
(d) What is the instantaneous rate of change in your annual fuel costs when x=5?
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6.
7.
8.
9.(a)
(b)
(c)
10(a)
(b)
(c)
(d) /(e) /
11.
Find an equation that relates the variables. /Differentiate both sides implicitly with respect to time “t”. You will need the product rule. /
Substitute in the “when moment” information. Note that if the area is not changing, its rate of change is 0. /
Solve for . /
Conclude with a sentence. / The width is decreasing at a rate of .
12.
Draw a sketch of the “whenever” situation showing all variables. /Find an equation that relates the variables. /
Use similar triangles to find a relationship between r and h. Solve for r and substitute into . The question asks us to find the rate at which the water level is rising, so we try to obtain an expression for the volume in terms of only the height of the water. /
Differentiate both sides implicitly with respect to time “t”. /
Substitute in the “when moment” information. /
Solve for . /
Conclude with a sentence. / The water level is rising at a rate of when the depth of the water is 20 cm.
13.
Draw a sketch of the “whenever” situation showing all variables. /Find an equation that relates the variables. /
The radius is constant regardless of the height, r = 3m. The question asks us to find the rate at which the water level is rising, so we try to obtain an expression for the volume in terms of only the height of the water. /
Differentiate both sides implicitly with respect to time “t”. /
Substitute in the rate of volume change. /
Solve for . /
Conclude with a sentence. / The water level is rising at a rate of .