Question

The formula \(h=80 t-16 t^{2}\) gives the height above the ground at time \(t\) for a ball that is thrown straight up in the air with an initial velocity of 80 feet per second. a. Complete a table of values for the following input values: \(t=0,1,2,3,4,\) and 5 seconds. b. Plot the input/output pairs from the table in part a on the coordinate system, and connect them in a smooth curve. c. Use the graph to estimate how high above the ground the ball will get. d. From the graph, when will the ball be 100 feet above the ground? e. When will the ball hit the ground? Give a reason for your answer.

Short answer

Based on the given projectile motion problem with the formula \(h=80 t-16 t^{2}\), follow these steps: Step 1: Calculate height values for given times - Plug in each given time value (\(t\)) into the formula to calculate the corresponding height (\(h\)). Step 2: Plot the points on the coordinate system - Use the table of values from Step 1 to plot the points on the coordinate system and draw a smooth curve. Step 3: Estimate the maximum height - Identify the highest point on the curve to approximate the maximum height the ball will reach. Step 4: Determine the time for the height of 100 feet - Locate the point(s) on the graph where the y-coordinate is 100 feet and read the corresponding x-coordinate, which is the time value. Step 5: Determine when the ball will hit the ground - Find the point on the graph where the height is 0 (the ball hits the ground) and read the corresponding time (x-coordinate) at which this occurs.

Step-by-step solution

Calculate height values for given times

To complete the table of values, we must find the height (\(h\)) for each of the given time values (\(t\)) using the formula \(h=80 t-16 t^{2}\). Calculate the height for each time and create a table of values. #b. Plot the input/output pairs on the coordinate system#

Plot the points on the coordinate system.

Using the table of values constructed in Step 1, plot each input (time) and output (height) on the coordinate system and connect the points in a smooth curve to represent the graph of the projectile motion. #c. Estimate maximum height from the graph#

Estimate the maximum height

Inspect the graph and identify the highest point of the smooth curve. This will give us an approximate maximum height the ball will reach above the ground. #d. Find the time it will take for the ball to be 100 feet above the ground#

Determine the time for the height of 100 feet

Examine the graph and locate the point(s) where the height (y-coordinate) is equal to 100 feet. Read the time value (x-coordinate) at which this occurs. #e. Determine when the ball will hit the ground#

Determine when the ball will hit the ground

Observe the graph and find the time at which the height reaches 0 (i.e. when the ball hits the ground). This is the time the ball will take to hit the ground.