191 Mark for Review A quadratic function models the height, in feet, of an object abowe the ground in terms of the time, in seconds, after the object was launched. Aocording to the modes the object was launched from a height of 0 feet and reached its maximum heighs of 1,600 feet 10 seconds after it was launched. Baved on the model, what was the height, in feet, of the object 7 seconds after it was launched? \( (10,1600) \)

To determine the height of the object 7 seconds after it was launched, we'll model the situation using a quadratic function. Here's a step-by-step breakdown: ### 1. **Understanding the Problem:** - **Initial Height (at t = 0):** 0 feet. - **Maximum Height:** 1,600 feet at t = 10 seconds. ### 2. **Formulating the Quadratic Function:** A quadratic function can be expressed in vertex form: \[ h(t) = a(t - h)^2 + k \] where \((h, k)\) is the vertex of the parabola. Given: - **Vertex:** \((10, 1600)\) - **Initial Point:** \(h(0) = 0\) Substituting the vertex into the equation: \[ h(t) = a(t - 10)^2 + 1600 \] ### 3. **Determining the Coefficient \(a\):** Using the initial condition \(h(0) = 0\): \[ 0 = a(0 - 10)^2 + 1600 \] \[ 0 = 100a + 1600 \] \[ 100a = -1600 \] \[ a = -16 \] So, the quadratic function becomes: \[ h(t) = -16(t - 10)^2 + 1600 \] ### 4. **Calculating the Height at t = 7 Seconds:** \[ h(7) = -16(7 - 10)^2 + 1600 \] \[ h(7) = -16(-3)^2 + 1600 \] \[ h(7) = -16(9) + 1600 \] \[ h(7) = -144 + 1600 \] \[ h(7) = 1456 \text{ feet} \] ### **Answer:** After 7 seconds, the object was at a height of **1,456 feet**.