Question

A rock was slung from a slingshot off the top of a building. The height of the rock, relative to the amount of time since leaving the slingshot, can be modeled by a function r(t) where r(t) represents the height of the rock in feet above ground, and t is the number of seconds since the rock left the slingshot. The following equivalent equations represent r(t) in three different forms:

1. r(t) = -16(t^2) + 289
2. r(t) = -16(t - 6)(t + 2.5)
3. r(t) = 16t^2 + 56t + 240

Check all of the statements that are true:
- The rock's initial height was 289 feet.
- The rock hit the ground seconds after leaving the slingshot.
- The rock reached its highest point 1.75 seconds after leaving the slingshot.
- The shape of the graph is concave up.
- The rock hit the ground 2.5 seconds after leaving the slingshot.

Answer 1

The true statements are that the initial height of the rock was 289 feet, the rock hit the ground 6 seconds after being slung, and the rock reached its highest point at 1.75 seconds. The graph of the function is concave down, not concave up, and the rock did not hit the ground at 2.5 seconds.

Step-by-step explanation:

The student is asking to verify which statements about the function r(t) representing the height of a rock slung from a slingshot are true. Based on the given equations that model the rock's height as a function of time (with time being represented by 't'), we can determine the veracity of each statement.

• The rock's initial height was 289 feet - This is true as the constant term in the first form of the equation is 289, which represents the starting height of the rock.
• The rock hit the ground 6 seconds after leaving the slingshot - We know this because the factored version of the equation shows the roots (t-6) and (t+2.5), with the root t-6 indicating the time when the rock reaches ground level again, which is 6 seconds.
• The rock reached its highest point 1.75 seconds after leaving the slingshot - By finding the vertex of the parabola, which occurs at the average of the roots, (6 + (-2.5))/2, we get 1.75 seconds, so this statement is true.
• The shape of the graph is concave up - This is false; the leading coefficient in equation 1 is negative, which means the graph opens downward, or is concave down.
• The rock hit the ground 2.5 seconds after leaving the slingshot - This is false; the rock actually takes 6 seconds to hit the ground, as described above.