The linear model represents the height, f(x), of a water balloon thrown off the roof of a building over time, x, measured in seconds:
A linear model with ordered pairs at 0, 60 and 2, 75 and 4, 75 and 6, 40 and 8, 20 and 10, 0 and 12, 0 and 14, 0. The x axis is labeled Time in seconds, and the y axis is labeled Height in feet.
Part A: During what interval(s) of the domain is the water balloon's height increasing? (2 points)
Part B: During what interval(s) of the domain is the water balloon's height staying the same? (2 points)
Part C: During what interval(s) of the domain is the water balloon's height decreasing the fastest? Use complete sentences to support your answer. (3 points)
Part D: Use the constraints of the real-world situation to predict the height of the water balloon at 16 seconds. Use complete sentences to support your answer. (3 points)
write this in 4 sentences
Part A: The water balloon's height is increasing during the interval from 0 to 2 seconds. This can be observed from the ordered pairs where the height goes from 60 feet to 75 feet.
Part B: The water balloon's height is staying the same during the interval from 2 to 4 seconds. This can be seen in the ordered pair (2, 75) and (4, 75) where the height remains constant at 75 feet.
Part C: The water balloon's height is decreasing the fastest during the interval from 6 to 8 seconds. This conclusion is drawn from the ordered pairs (6, 40) and (8, 20), where the height drops from 40 feet to 20 feet over a span of 2 seconds.
Part D: Based on the linear model, we can predict that the height of the water balloon at 16 seconds will be 0 feet. This prediction is based on the trend observed in the given data: as time increases, the height of the water balloon decreases until it reaches 0 feet at 16 seconds.
The linear function f(x) = 0.5x + 80 represents the average test score in your math class, where x is the number of the test taken. The linear function g(x) represents the average test score in your science class, where x is the number of the test taken.
x g(x)
1 81
2 83
3 85
Part A: Determine the test average for your math class after completing test 2. (2 points)
Part B: Determine the test average for your science class after completing test 2. (2 points)
Part C: Which class had a higher average after completing test 4? Show work to support your answer. (6 points)
Part A: To determine the test average for your math class after completing test 2, plug in x = 2 into the equation f(x) = 0.5x + 80.
f(2) = 0.5(2) + 80
f(2) = 81
So, the test average for your math class after completing test 2 is 81.
Part B: To determine the test average for your science class after completing test 2, we look at the given information. The average for test 1 is 81, and the average for test 2 is 83. Therefore, the test average for your science class after completing test 2 is 83.
Part C: To determine which class had a higher average after completing test 4, we need to find the test average for each class after completing test 4.
For the math class, we use the same equation f(x) = 0.5x + 80 and plug in x = 4:
f(4) = 0.5(4) + 80
f(4) = 82
For the science class, we need to find the average for test 3 and test 4. According to the given information, the average for test 3 is 85. To find the average for test 4, we assume that the trend continues with a constant increase of 2.
Test 4 average = Test 3 average + increase
Test 4 average = 85 + 2
Test 4 average = 87
Comparing the averages, we can see that the math class had an average of 82 after completing test 4, while the science class had an average of 87 after completing test 4. Therefore, the science class had a higher average after completing test 4.
The table of values represents a linear function g(x), where x is the number of days that have passed and g(x) is the balance in the bank account:
x g(x)
0 $1,500
2 $1,350
4 $1,200
Part A: Find and interpret the slope of the function. (3 points)
Part B: Write the equation of the line in point-slope, slope-intercept, and standard forms. (3 points)
Part C: Write the equation of the line using function notation. (2 points)
Part D: What is the balance in the bank account after 5 days? (2 points)
Part A: To find the slope, we can use the formula: slope = (change in y)/(change in x).
Using the values from the table, we can calculate the change in y and the change in x:
(change in y) = $1,200 - $1,500 = -$300
(change in x) = 4 - 0 = 4
Now we can calculate the slope:
slope = (-$300)/(4) = -$75
Interpretation: The slope of the function is -$75, which means that for every increase of 1 in the number of days, the balance in the bank account decreases by $75.
Part B:
Point-slope form: We can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Using the point (0, $1,500) and the slope -$75, the equation in point-slope form is:
g(x) - $1,500 = -$75(x - 0)
Slope-intercept form: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
Substituting the values of the slope and the y-intercept (from the point (0, $1,500)) into the equation:
g(x) = -$75x + $1,500
Standard form: The standard form of a linear equation is Ax + By = C, where A, B, and C are constants.
Converting the equation from slope-intercept form to standard form:
$75x + g(x) = $1,500
Part C: Using function notation, the equation of the line can be written as:
g(x) = -$75x + $1,500
Part D: To find the balance in the bank account after 5 days, we can substitute x = 5 into the equation:
g(5) = -$75(5) + $1,500 = $1,125
Therefore, the balance in the bank account after 5 days is $1,125.
Part A: The water balloon's height is increasing during the interval from 0 to 2 seconds. This is because the height at 0 seconds is 60 feet, and at 2 seconds, it increases to 75 feet.
Part B: The water balloon's height is staying the same during the intervals from 2 to 4 seconds and from 10 to 14 seconds. This is evident from the ordered pairs (2, 75), (4, 75), (10, 0), (12, 0), and (14, 0), where the height remains constant.
Part C: The water balloon's height is decreasing the fastest during the interval from 4 to 6 seconds. Within this interval, the height drops from 75 feet at 4 seconds to 40 feet at 6 seconds, indicating a significant rate of descent.
Part D: Based on the given linear model, we can predict the height of the water balloon at 16 seconds to be 0 feet. This estimate is derived from the fact that the height remains constant at 0 feet from 12 to 14 seconds, and the linear trend suggests a continued downward trend beyond the given data points.
To determine the intervals during which the water balloon's height is increasing or staying the same, we need to analyze the slope of the linear model. The slope represents the rate of change in height with respect to time. When the slope is positive, the height is increasing, while a zero slope indicates the height remains constant. To find the intervals with a positive slope, we compare the height at successive time points. We observe that from 0 to 2 seconds, 2 to 4 seconds, and then 8 to 10 seconds, the height increases. Thus, the water balloon's height is increasing during the intervals 0 to 2, 2 to 4, and 8 to 10 seconds.
To identify the intervals where the height remains the same, we look for consecutive time points with the same value for the height. In this case, from 4 to 6 seconds, the height remains constant at 75 feet. Therefore, the water balloon's height stays the same during the interval 4 to 6 seconds.
To determine the interval where the water balloon's height decreases the fastest, we compare the rate of change in height among all intervals. By examining the slopes of different intervals, we observe that the fastest decrease in height occurs between 6 and 8 seconds. During this interval, the height drops from 40 to 20 feet, resulting in a steeper slope compared to other intervals. Hence, the water balloon's height decreases the fastest between 6 and 8 seconds.
Lastly, to predict the height of the water balloon at 16 seconds, we must assume that the linear model accurately represents the balloon's trajectory after 14 seconds. Since the height remains zero from 10 to 14 seconds, we can infer that the height will continue to be zero at 16 seconds. However, it's essential to note that this prediction is based solely on the provided data and assumptions made for the linear model.