1.) The cash flow per share for the Timerland Co. was $0.18 in 1995 and $4.04 in 2003. Write a liner equation that gives the cash flow per share in terms of the year. Let t=5 represent 1995. Then predict the cash flows for the years 2008 and 2010.
5.) The University of Florida had enrollments of 36,531 students in 1990 and 48,673 students in 2003.
a.) What was the average annual change in enrollment from 1990 to 2003?
b) Use the average annual change in enrollment to estimate the enrollments in 1994, 1998, & 2002.
c) Write the equation of a line that represents the given data. What is its slope? Interpret the slope in the context of the problem.
d) Using the results of parts (a-c)
Write a short paragraph discussing the concepts of slope and average rate of change?
In 8 years, the share price rose by 3.86, making the rate of change (slope) 3.86/8 = 0.4825 per year.
So, since we started measuring at the $0.18 price, at t=5, the price p(t) is
p-0.18 = 0.4825(t-5)
Now you can find p(t) for other values of t.
The other problem is worked in similar manner.
1.) To find a linear equation that gives the cash flow per share in terms of the year, we first need to identify the slope and y-intercept. We are given the cash flow per share values for 1995 and 2003.
Let's assume that t represents the number of years since 1995. In 1995, t = 0, and in 2003, t = 8.
So, we have two points (0, 0.18) and (8, 4.04).
To find the slope, we use the formula:
slope = (y2 - y1) / (x2 - x1)
= (4.04 - 0.18) / (8 - 0)
= 3.86 / 8
= 0.4825
Now that we have the slope, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the point (0, 0.18), we have:
y - 0.18 = 0.4825(x - 0)
y = 0.4825x + 0.18
Now, let's predict the cash flows for the years 2008 and 2010.
For 2008, t = 13 (since 2008 - 1995 = 13), substituting this into the equation:
y = 0.4825(13) + 0.18
y = 6.2675 + 0.18
y = 6.4475
So the predicted cash flow per share for 2008 is $6.4475.
For 2010, t = 15 (since 2010 - 1995 = 15), substituting this into the equation:
y = 0.4825(15) + 0.18
y = 7.2375 + 0.18
y = 7.4175
So the predicted cash flow per share for 2010 is $7.4175.
5.)
a.) To find the average annual change in enrollment from 1990 to 2003, we need to calculate the difference in enrollments and divide it by the number of years.
Change in enrollment = 48,673 - 36,531 = 12,142
Number of years = 2003 - 1990 = 13
Average annual change = Change in enrollment / Number of years
= 12,142 / 13
= 934.77 (approximately)
The average annual change in enrollment from 1990 to 2003 is approximately 934.77 students.
b) We can use the average annual change in enrollment to estimate the enrollments in 1994, 1998, and 2002.
For 1994, we add 4 years to 1990:
1990 + 4 = 1994
Predicted enrollment in 1994 = Previous enrollment + (Average annual change * Number of years)
= 36,531 + (934.77 * 4)
= 36,531 + 3,739.08
= 40,270 (approximately)
So the estimated enrollment in 1994 is approximately 40,270 students.
Similarly, for 1998:
1990 + 8 = 1998
Predicted enrollment in 1998 = Previous enrollment + (Average annual change * Number of years)
= 36,531 + (934.77 * 8)
= 36,531 + 7,478.16
= 44,010 (approximately)
So the estimated enrollment in 1998 is approximately 44,010 students.
And for 2002:
1990 + 12 = 2002
Predicted enrollment in 2002 = Previous enrollment + (Average annual change * Number of years)
= 36,531 + (934.77 * 12)
= 36,531 + 11,217.24
= 47,748 (approximately)
So the estimated enrollment in 2002 is approximately 47,748 students.
c) To write the equation of a line that represents the given data, we need to identify the slope and y-intercept. We have two points, (1990, 36,531) and (2003, 48,673).
Using the formula for slope:
slope = (y2 - y1) / (x2 - x1)
= (48,673 - 36,531) / (2003 - 1990)
= 12,142 / 13
= 934.77 (approximately)
Now we can use the point-slope form of a linear equation with the point (1990, 36,531):
y - y1 = m(x - x1)
y - 36,531 = 934.77(x - 1990)
y = 934.77x - 1,755,666.3
The equation of the line that represents the given data is y = 934.77x - 1,755,666.3. The slope of this line is 934.77.
Interpreting the slope in the context of the problem, for each year increase in the independent variable (year), the enrollment is estimated to increase by approximately 934.77 students.
d) The concepts of slope and average rate of change are closely related. Slope measures the steepness or incline of a line, while average rate of change represents the average amount of change in one variable per unit change in another variable over a specific interval.
In the context of the cash flow per share problem, the slope of the linear equation represents how the cash flow per share changes over time. It quantifies the average increase in cash flow per share for each additional year that passes.
In the context of the enrollment problem, the slope of the line represents the average annual change in enrollment between 1990 and 2003. It quantifies the average increase in the number of students enrolled per year.
Both slope and average rate of change allow us to quantify the change in one variable with respect to another and can be used to make predictions or estimations for values not explicitly given in the data set.