Find Vertex Time: To find the time it takes for the t-shirt to reach its maximum height, we need to find the vertex of the parabola represented by the quadratic function $h(t) = -16t^2 + 72t + 5$ . The time at which the maximum height is reached is given by the $t$ -coordinate of the vertex of the parabola. The formula to find the $t$ -coordinate of the vertex is $-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients from the quadratic equation in the form of $h(t) = at^2 + bt + c$ .
Calculate Vertex Time: In our equation, $a = -16$ and $b = 72$ . We substitute these values into the formula to find the $t$ -coordinate of the vertex: $t = -b/(2a) = -72/(2 \times -16) = -72 / -32 = 2.25$ .
Maximum Height Calculation: Therefore, it will take $2.25$ seconds for the t-shirt to reach its maximum height.
Substitute Time in Equation: To find the maximum height, we substitute the time back into the original equation. So we calculate $h(2.25) = -16(2.25)^2 + 72(2.25) + 5$ .
Calculate $(2.25)^2$ : First, we calculate $(2.25)^2 = 5.0625$ .
Calculate $-16 \times 5.0625$ : Next, we calculate $-16 \times 5.0625 = -81$ .
Calculate $72 \times 2.25$ : Then, we calculate $72 \times 2.25 = 162$ .
Add All Terms for Height: Now, we add all the terms together: $h(2.25) = -81 + 162 + 5 = 86$ .
Catch T-shirt at $35.02$ feet: The maximum height reached by the t-shirt is $86$ feet.
Set up Equation for $t$ : For part (c), we are asked to use a graphing calculator, which is not possible in this text format. However, we can still solve for the time it takes for someone to catch the t-shirt at $35.02$ feet without a graphing calculator by setting the height function equal to $35.02$ and solving for $t$ .
Quadratic Formula for $t$ : We set up the equation $-16t^2 + 72t + 5 = 35.02$ .
Calculate Discriminant: To solve for $t$ , we first subtract $35.02$ from both sides to get $-16t^2 + 72t - 30.02 = 0$ .
Square Root of Discriminant: This is a quadratic equation in standard form, and we can use the quadratic formula to solve for $t$ : $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = -16$ , $b = 72$ , and $c = -30.02$ .
Use Quadratic Formula for $t$ : First, we calculate the discriminant: $b^2 - 4ac = 72^2 - 4(-16)(-30.02)$ .
Calculate Two Possible Times: Calculating the discriminant gives us: $5184 - 1920.64 = 3263.36$ .
Total Time from Launch: Now we take the square root of the discriminant: $\sqrt{3263.36} \approx 57.14$ .
Total Time from Launch: Now we take the square root of the discriminant: $\sqrt{3263.36} \approx 57.14$ .We can now use the quadratic formula to find the two possible values for $t$ : $t = (72 \pm 57.14) / -32$ .
Total Time from Launch: Now we take the square root of the discriminant: $\sqrt{3263.36} \approx 57.14$ .We can now use the quadratic formula to find the two possible values for $t$ : $t = (72 \pm 57.14) / -32$ .Calculating the two possible times gives us: $t = (72 + 57.14) / -32 \approx -4.04$ (which is not possible since time cannot be negative) and $t = (72 - 57.14) / -32 \approx 0.464$ seconds.
Total Time from Launch: Now we take the square root of the discriminant: $\sqrt{3263.36} \approx 57.14$ .We can now use the quadratic formula to find the two possible values for $t$ : $t = (72 \pm 57.14) / -32$ .Calculating the two possible times gives us: $t = (72 + 57.14) / -32 \approx -4.04$ (which is not possible since time cannot be negative) and $t = (72 - 57.14) / -32 \approx 0.464$ seconds.The time it took for someone to catch the t-shirt is approximately $0.464$ seconds after the maximum height is reached. Since the maximum height is reached at $2.25$ seconds, we add this time to find the total time from launch to catch: $2.25 + 0.464 \approx 2.714$ seconds.