Length and Cliff
d
4.4 The Fisherman – Part 1
A fisherman is standing on the edge of a cliff overlooking a flowing river. The edge of the cliff is 8 feet above the level of the water. The tip of his fishing rod is directly above the edge of the cliff and is 7 feet above the cliff. He lowers his fishing line so that it enters the water at the base of the cliff. The length of his fishing line, l, which is measured from the tip of his fishing rod to the place his line enters the water, and the angle between his line and the water, θ, are changing as the river carries his line downstream. The horizontal distance, r, from the edge of the cliff to the place his line enters the water increases at a rate of 1ft/2sec as the river carries it downstream.
1. Draw a diagram of the situation, using l, r and θ.
2. Complete the table below where A represents the area of the triangle formed by the height of the line, the distance from the base of the cliff, and the length of the line. t, in sec r, in ft
A, in ft2
4
2
15
16
8
60
20
10
75
3. What is an equation for r(t)? r = .5t
4. What is an equation for A(t)? A = 7.5(.5t)
5. Sketch the graphs of r(t) and A(t) on the same axes. Indicate which is which.
6. In any equation y = mx + b, the constant m is used to represent the slope. In a linear equation, the slope is equivalent to Δy/Δx (the average rate of change of y with respect to x) and is equivalent to dy/dx (the instantaneous rate of change of y with respect to x). What is the rate of change in the distance from the edge of the cliff to the place the fisherman’s line enters the water with respect to time? Use the equation for r(t) you found in #3 and include correct units. m = ½ ft/sec Δr/Δt = 2-8 ft/4-16 sec = -6 ft/-12 sec = ½ ft/sec dr/dt = 2/4 = ½ ft/sec
7. What is the rate of change in the area of the triangle with respect to time? Use the equation for A(t) you found in #4 and include correct units. m = 3.75 ft²/sec
4.4 The Fisherman – Part 1
A fisherman is standing on the edge of a cliff overlooking a flowing river. The edge of the cliff is 8 feet above the level of the water. The tip of his fishing rod is directly above the edge of the cliff and is 7 feet above the cliff. He lowers his fishing line so that it enters the water at the base of the cliff. The length of his fishing line, l, which is measured from the tip of his fishing rod to the place his line enters the water, and the angle between his line and the water, θ, are changing as the river carries his line downstream. The horizontal distance, r, from the edge of the cliff to the place his line enters the water increases at a rate of 1ft/2sec as the river carries it downstream.
1. Draw a diagram of the situation, using l, r and θ.
2. Complete the table below where A represents the area of the triangle formed by the height of the line, the distance from the base of the cliff, and the length of the line. t, in sec r, in ft
A, in ft2
4
2
15
16
8
60
20
10
75
3. What is an equation for r(t)? r = .5t
4. What is an equation for A(t)? A = 7.5(.5t)
5. Sketch the graphs of r(t) and A(t) on the same axes. Indicate which is which.
6. In any equation y = mx + b, the constant m is used to represent the slope. In a linear equation, the slope is equivalent to Δy/Δx (the average rate of change of y with respect to x) and is equivalent to dy/dx (the instantaneous rate of change of y with respect to x). What is the rate of change in the distance from the edge of the cliff to the place the fisherman’s line enters the water with respect to time? Use the equation for r(t) you found in #3 and include correct units. m = ½ ft/sec Δr/Δt = 2-8 ft/4-16 sec = -6 ft/-12 sec = ½ ft/sec dr/dt = 2/4 = ½ ft/sec
7. What is the rate of change in the area of the triangle with respect to time? Use the equation for A(t) you found in #4 and include correct units. m = 3.75 ft²/sec