`d(t) = -16t^2+40t +24`
A swimmer dives from a diving board that is 24 feet above the water. The distance, in feet, that the diver travels after t seconds have elapsed is given by the function above. What is the maximum height above the water, in feet, the swimmer reaches during the dive?

Answer

To find the maximum height above the water that the swimmer reaches during the dive, we can follow these steps:

### Step 1: Understand the function
The distance function given is:
\[ d(t) = -16t^2 + 40t + 24 \]
This function represents the distance traveled by the diver after \( t \) seconds, where the diver starts from a height of 24 feet above the water.

### Step 2: Find the vertex of the parabola
Since the function is a quadratic equation in the form \( d(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 40 \), and \( c = 24 \), we can find the time \( t \) at which the maximum height occurs using the vertex formula:
\[ t = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{40}{2 \times -16} = \frac{40}{32} = \frac{5}{4} \]

### Step 3: Calculate the maximum height
Now that we have the time \( t = \frac{5}{4} \), we can substitute this back into the distance function to find the maximum height:
\[ d\left(\frac{5}{4}\right) = -16\left(\frac{5}{4}\right)^2 + 40\left(\frac{5}{4}\right) + 24 \]

Calculating each term:
1. Calculate \( \left(\frac{5}{4}\right)^2 = \frac{25}{16} \)
2. Calculate \( -16 \times \frac{25}{16} = -25 \)
3. Calculate \( 40 \times \frac{5}{4} = 50 \)

Now substituting these values into the equation:
\[ d\left(\frac{5}{4}\right) = -25 + 50 + 24 \]

### Step 4: Simplify the expression
Now, simplifying the expression:
\[ d\left(\frac{5}{4}\right) = 25 + 24 = 49 \]

### Conclusion
Thus, the maximum height above the water that the swimmer reaches is:
\[ \boxed{49} \text{ feet} \]

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