25. the sears tower, at 1,451 feet, is one of the tallest structures in the united states. a penny is thrown…

25. the sears tower, at 1,451 feet, is one of the tallest structures in the united states. a penny is thrown from the top of the tower. the height, $h$, of the penny is recorded after each second, $t$, in the table. using an equation to model the data, find the approximate height of the penny after 7 seconds.\n\n| $t$ | $h$ |\n| :--- | :--- |\n| 0 | 1451 |\n| 1 | 1470 |\n| 2 | 1457 |\n| 3 | 1412 |\n| 4 | 1335 |\n\na. 907 feet\nb. 912 feet\nc. 923 feet\nd. 928 feet

25. the sears tower, at 1,451 feet, is one of the tallest structures in the united states. a penny is thrown from the top of the tower. the height, $h$, of the penny is recorded after each second, $t$, in the table. using an equation to model the data, find the approximate height of the penny after 7 seconds.\n\n| $t$ | $h$ |\n| :--- | :--- |\n| 0 | 1451 |\n| 1 | 1470 |\n| 2 | 1457 |\n| 3 | 1412 |\n| 4 | 1335 |\n\na. 907 feet\nb. 912 feet\nc. 923 feet\nd. 928 feet

Answer

Explanation:

Step1: Identify the projectile motion model

The height $h$ of an object thrown vertically is modeled by a quadratic equation $h(t) = at^2 + bt + c$.

Step2: Determine the initial height constant

At $t = 0$, $h = 1451$, so $c = 1451$.

Step3: Use gravity constant for feet

In physics, the acceleration due to gravity in feet per second squared is approximately $32$, making the leading coefficient $a = -16$.

Step4: Solve for initial velocity

Using the data point $(1, 1470)$: $$1470 = -16(1)^2 + b(1) + 1451$$ $$1470 = -16 + b + 1451$$ $$1470 = b + 1435 \implies b = 35$$

Step5: Verify the model with another point

Check $(2, 1457)$: $$h(2) = -16(2)^2 + 35(2) + 1451 = -64 + 70 + 1451 = 1457$$ The model is $h(t) = -16t^2 + 35t + 1451$.

Step6: Calculate height at seven seconds

Substitute $t = 7$ into the equation: $$h(7) = -16(7)^2 + 35(7) + 1451$$ $$h(7) = -16(49) + 245 + 1451$$ $$h(7) = -784 + 245 + 1451$$ $$h(7) = 912$$

Answer:

B. 912 feet