A model rocket is launched vertically into the air such that its height at any time, t, is given by the function `h(t)=-16t^(2)+80t+10`. What is the maximum height attained by the model rocket?

Options

A. `140`
B. `110`
C. `85`
D. `10`

Answer

To find the maximum height attained by the model rocket, we will follow these steps:

### Step 1: Identify the height function
The height of the rocket at any time \( t \) is given by the function:
\[ h(t) = -16t^2 + 80t + 10 \]

### Step 2: Differentiate the height function
To find the maximum height, we need to find the critical points of the function. We do this by differentiating \( h(t) \) with respect to \( t \):
\[ h'(t) = \frac{d}{dt}(-16t^2 + 80t + 10) = -32t + 80 \]

### Step 3: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero:
\[ -32t + 80 = 0 \]

### Step 4: Solve for \( t \)
Now, we solve for \( t \):
\[ -32t = -80 \\ t = \frac{80}{32} = 2.5 \]

### Step 5: Find the maximum height
Now that we have the time \( t = 2.5 \), we will substitute this back into the height function to find the maximum height:
\[ h(2.5) = -16(2.5)^2 + 80(2.5) + 10 \]

Calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Now substituting back:
\[ h(2.5) = -16(6.25) + 80(2.5) + 10 \\ = -100 + 200 + 10 \\ = 110 \]

### Conclusion
The maximum height attained by the model rocket is:
\[ \boxed{110} \]