Ingrid is docking a motorboat. She turns off the power and lets the boat coast toward the dock. The distance, d, in meters, between the boat and the dock as a function of time, t, in seconds, is given by d(t)=-7t+70/t + 4,
0 ≤ t ≤ 10
a) What is the average velocity of the boat during the time interval from when Ingrid turns the boat off to when it meets the dock?
b) Determine the velocity of the boat when Ingrid turns off the power, to one decimal place.
c) Determine the velocity of the boat when it meets the dock, to one decimal place.
d) Sketch a graph of the function. On the same set of axes, sketch a graph of the speed of the boat in relation to the time.
a) The average velocity of the boat during the time interval from when Ingrid turns the boat off to when it meets the dock can be found by finding the total change in position divided by the total time elapsed.
To find the total change in position, we need to evaluate d(t) at the start and end of the interval:
d(0) = -7(0) + 70/0 + 4 (Note: we cannot divide by 0, so we will substitute a very small value, such as 0.001)
d(10) = -7(10) + 70/10 + 4
Using these values, we can calculate the change in position:
Change in position = d(10) - d(0)
To find the total time elapsed, we subtract the starting time from the ending time:
Total time elapsed = 10 seconds - 0 seconds
Finally, we can calculate the average velocity:
Average velocity = Change in position / Total time elapsed
b) The velocity of the boat when Ingrid turns off the power can be found by evaluating the derivative of the position function at that specific time, t = 0.
To find the derivative, we differentiate the position function with respect to t:
d'(t) = -7 - 70/t^2
Evaluate d'(0) to find the velocity at t = 0.
c) The velocity of the boat when it meets the dock can be found by evaluating the derivative of the position function at that specific time, t = 10.
To find the derivative, we differentiate the position function with respect to t:
d'(t) = -7 - 70/t^2
Evaluate d'(10) to find the velocity at t = 10.
d) To sketch a graph of the function, plot the values of d(t) for t from 0 to 10. Then, on the same set of axes, plot the values of the speed of the boat (absolute value of the velocity) in relation to time, using the derivatives calculated in parts b and c.
a) To find the average velocity of the boat during the time interval from when Ingrid turns the boat off to when it meets the dock, we need to find the change in distance and the change in time.
The initial distance when Ingrid turns off the power is given by d(0) = -7(0) + 70/0 + 4, which is undefined since we cannot divide by 0.
However, we can consider an infinitely small positive time value, say t = ε, before turning off the power. At this time, the distance from the dock is given by d(ε) = -7ε + 70/ε + 4.
The final distance when the boat meets the dock is given by d(10) = -7(10) + 70/10 + 4 = -70 + 7 + 4 = -59.
The change in distance is (-59) - d(ε) = -59 - (-7ε + 70/ε + 4) = -59 + 7ε - 70/ε - 4.
The change in time is 10 - ε.
Therefore, the average velocity is (change in distance) / (change in time):
Average velocity = [(-59 + 7ε - 70/ε - 4) / (10 - ε)]
b) To determine the velocity of the boat when Ingrid turns off the power, we need to find the instantaneous velocity at t = 0.
The velocity is given by the derivative of the distance function:
Velocity = d'(t) = (-7) - (70/t^2) = -7 - 70/t^2.
At t = 0, the velocity of the boat is undefined since we cannot divide by 0.
c) To determine the velocity of the boat when it meets the dock, we need to find the instantaneous velocity at t = 10.
The velocity is given by the derivative of the distance function:
Velocity = d'(t) = (-7) - (70/t^2) = -7 - 70/(10^2) = -7 - 70/100 = -7 - 0.7 = -7.7.
Therefore, the velocity of the boat when it meets the dock is -7.7 m/s.
d) To sketch the graph of the function, we can plot a few points and connect them to form a smooth curve. Here are a few points along with their corresponding distances:
t = 0, d(0) = -7(0) + 70/0 + 4 (undefined)
t = 1, d(1) = -7(1) + 70/1 + 4 = -7 + 70 + 4 = 67
t = 2, d(2) = -7(2) + 70/2 + 4 = -14 + 35 + 4 = 25
t = 5, d(5) = -7(5) + 70/5 + 4 = -35 + 14 + 4 = -17
t = 10, d(10) = -7(10) + 70/10 + 4 = -70 + 7 + 4 = -59
To sketch the graph of the speed of the boat in relation to time, we can calculate the absolute value of the velocity at each time point and plot those values on the same set of axes.
To answer these questions, we need to analyze the given function d(t) = -7t + 70/t + 4 and understand its properties. Let's break it down step by step.
a) The average velocity of the boat is the total displacement divided by the total time. In this case, the displacement is the change in the distance between the boat and the dock, which is d(10) - d(0). The total time is 10 seconds. Substituting the values into the equation, we have:
Average velocity = (d(10) - d(0)) / 10
To find d(10), substitute t = 10 into the function:
d(10) = -7(10) + 70/(10) + 4
Simplifying, we get d(10) = -70 + 7 + 4 = -59
To find d(0), substitute t = 0 into the function:
d(0) = -7(0) + 70/(0) + 4
Note that dividing by zero is undefined, so the term 70/(0) is not valid. However, since we are interested in the average velocity from the moment Ingrid turns off the power, we do not need to calculate the initial position. Therefore, we can disregard the term d(0) in this case.
Substituting the values into the equation for average velocity:
Average velocity = (-59 - d(0)) / 10
b) The velocity of the boat when Ingrid turns off the power is instantaneous velocity and can be determined by taking the derivative of the function with respect to t. So, find the derivative of d(t):
d'(t) = -7 - 70/(t^2)
Substituting t = 0 (the moment Ingrid turns off the power):
Instantaneous velocity = -7 - 70/(0^2)
= -7
So, the velocity of the boat when Ingrid turns off the power is -7 meters per second.
c) To determine the velocity of the boat when it meets the dock, we need to find the limit of the function as t approaches 0. Calculating the limit:
Lim (t -> 0) (d(t)) = Lim (t -> 0) (-7t + 70/t + 4)
As t approaches 0, the first term (-7t) approaches 0, and the last term (4) remains unchanged. However, the middle term (70/t) approaches positive or negative infinity, depending on the sign of t. Since we are approaching from the positive side (0+), the limit is positive infinity.
Therefore, the velocity of the boat when it meets the dock is positive infinity.
d) To sketch the graph of the function, plot points for various values of t within the given interval [0, 10], and connect them with a smooth curve. Remember to include the y-intercept, which is the point (0, 4), and the x-intercept, which can be found by solving the equation d(t) = 0. This gives:
-7t + 70/t + 4 = 0
-7t^2 + 4t + 70 = 0
Solve this quadratic equation to find the x-intercept(s). Once you have plotted the graph, you can sketch the graph of speed in relation to time by taking the absolute value of the derivative of the function |d'(t)| = |-7 - 70/(t^2)|. This will give the magnitude of velocity at each point, representing speed.