I don't have a graphing calc so I need help with this one. A batter hits a fly ball. A scout in the stands makes the following observations. There is a table with time (seconds) and height (feet) in correspondence to each other time is .75,1.5,2,3.25,4.75 and height is 77,133,160,187,194,169 I am now supposed to a circle the type of function that best models this data either linear, exponential, quadratic, and other the answer is quadratic. B I can't figure out with out a GraphCalc. (b) Use a graphing calculator to preform the regression for the best-fit equation. Write the resulting equation below, rounding to the nearest hundredth. (c) Determine the initial velocity of the baseball and the height of the ball when hit. Round to the nearest hundredth. (d) Calculate how many seconds an outfielder has to position himself for the catch if he intends to catch the ball 6ft above the ground. Show your work and round to the nearest hundredth. Many thanks :)
If you graph the table of values and draw a line through the coordinates, the graph will look like a parabola. Thus the equation is quadratic. Since you don't have a graphing calculator, I will give you the resulting a/b/c values that it comes up with, but the rest is for you to figure out.
y=ax^2+bx+c
a= -15.90232286
b= 110.4844181
c= 3.015677435
To find the best-fit equation for the given data, you can use a graphing calculator that can perform regression analysis. Since you mentioned not having access to a graphing calculator, I will guide you through the steps using a free online graphing calculator called Desmos.
Step (a): Circling the type of function
To determine the type of function that best models the data, we can analyze the pattern in the given height values. By observing the data points, it suggests that the height of the ball changes with time in a curved manner. This kind of behavior is typically modeled by a quadratic function.
Step (b): Performing the regression analysis
1. Go to the website "desmos.com".
2. Click on the "Graphing Calculator" option from the menu.
3. In the input bar at the top, enter the pairs of numbers in the form (time, height). For example,
- (0.75, 77)
- (1.5, 133)
- (2, 160)
- (3.25, 187)
- (4.75, 194)
- (B, 169)
Note: Replace 'B' with the corresponding time value.
4. Once you have entered all the pairs, press enter.
Step (c): Determining the initial velocity and height of the ball when hit
To determine the initial velocity and height of the ball when hit, we need to find the equation of the quadratic function that best fits the data.
1. On the Desmos graph, click on the "+" button in the upper-left corner.
2. Select "Regression: Quadratic".
3. This will plot a quadratic curve that best fits the data.
4. Now, hover your mouse over the curve to view its equation. It should be in the form of y = ax² + bx + c.
- The coefficient 'a' represents the acceleration due to gravity.
- The coefficient 'b' corresponds to the initial velocity of the ball.
- The constant term 'c' represents the initial height of the ball when hit.
Note down the values of 'a', 'b', and 'c' from the equation.
Step (d): Calculating the time for the outfielder to position himself
To find the time an outfielder has to position himself for the catch when the ball is 6 feet above the ground, we can solve the quadratic equation.
1. Set the equation equal to 6: ax² + bx + c = 6.
2. Rearrange the equation to the quadratic form: ax² + bx + (c - 6) = 0.
3. Use the quadratic formula or the graphing calculator to find the two values of 'x'.
- These two values will represent the times when the ball is 6 feet above the ground.
4. Round your answer to the nearest hundredth.
I hope this explanation helps you in solving the problem. If you have any further questions, please feel free to ask.