Part A: Sam rented a boat at $225 for 2 days. If he rents the same boat for 5 days, he has to pay a total rent of $480.
Write an equation in the standard form to represent the total rent (y) that Sam has to pay for renting the boat for x days. (4 points)
Part B: Write the equation obtained in Part A using function notation.(2 points)
Part C: Describe the steps to graph the equation obtained above on the coordinate axes. Mention the labels on the axes and the intervals. (4 points)
PART A:
1. The curve of cost vs time is concave down.
2. This suggests a power curve of the form: a(time)^n = rent, where a and n are to be determined.
3. Write: at^n = r
4. Take nstural logarthm of both sides: ln(a) รท n[ln(t)] = ln(r)
5. Use given data:
r = 225 when t = 2 and
r = 480 when t = 5
6. Write resulting equations and solve:
ln(a) + 0.69t = 5.42
ln(a) + 1.62t = 6.17
Therefore a = 127.71 and n=0.82
7. The final result is:
127.71(t^0.82) = r
PART B:
r(t) = 127.71(t^0.82)
PART C:
The x-axis is time in days, the y axis is rent in hundreds of dollars.
Part A: To find the equation in the standard form, we can start by recognizing that the rent for 2 days is $225 and the rent for 5 days is $480. We can assume that the rent is increasing at a constant rate per day. Let's call this rate "r."
For 2 days, the rent is $225: 2r = 225
For 5 days, the rent is $480: 5r = 480
To find the value of "r," we can divide both equations by their respective number of days:
2r/2 = 225/2
r = 112.5
Now that we know the daily rate of rent, we can write the equation in the standard form:
y = rx
Substituting the value of "r" into the equation:
y = 112.5x
Part B: To write the equation obtained in Part A using function notation, we can replace "y" with "f(x)":
f(x) = 112.5x
Part C: To graph the equation obtained above, we can label the horizontal axis as "x" representing the number of days and the vertical axis as "f(x)" representing the total rent.
To determine the intervals for the axes, we need to consider the range of values for both "x" and "f(x)." Since we are not given any restrictions on the number of days and we are looking for the general equation, we can choose a reasonable range for the axes.
For the horizontal axis, we can choose intervals in increments of 1, representing each day from 0 to a suitable number of days. Let's choose 10 days.
For the vertical axis, we need to consider the range of possible total rents. Since we know that the daily rate of rent is $112.5, the total rent will increase as the number of days increases. Let's choose the maximum total rent as $600.
Therefore, we can label the horizontal axis as "x" in increments of 1, from 0 to 10, and the vertical axis as "f(x)" in increments of 100, from 0 to 600.
Plotting the points (0, 0), (2, 225), and (5, 480) on the graph, we can then draw a straight line connecting these points. This line represents the equation f(x) = 112.5x.