A projectile is thrown upward so that its distance above the ground after t seconds is 

h left parenthesis t right parenthesis equals negative 16 t squared plus 260 t.h(t)=−16t2+260t. 

After how many seconds does it reach its maximum​ height?

Question content area bottom

Part 1

A.

2626 

sec

B.

66 

sec

C.

88 

sec 

D.

19.519.5 

sec

To determine when the projectile reaches its maximum height, we need to find the vertex of the quadratic function \( h(t) = -16t^2 + 260t \). The vertex of a parabola described by the equation \( at^2 + bt + c \) can be found using the formula:

\[ t = -\frac{b}{2a} \]

In this case, the function is \( h(t) = -16t^2 + 260t \), where \( a = -16 \) and \( b = 260 \). Plugging these values into the formula, we get:

\[ t = -\frac{260}{2 \cdot -16} \]
\[ t = -\frac{260}{-32} \]
\[ t = \frac{260}{32} \]
\[ t = 8.125 \]

So, the projectile reaches its maximum height after 8.125 seconds. However, since none of the options exactly match 8.125 seconds, let's verify that our calculations are correct. 

Let's re-evaluate the given options:

- A. \( 26 \) sec
- B. \( 6 \) sec
- C. \( 8 \) sec
- D. \( 19.5 \) sec

Since 8.125 seconds is very close to 8 seconds, the closest and correct answer is:

- **C. 8 sec**

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How do you derive the formula for the vertex of a quadratic function?
2. What is the significance of the coefficients in a quadratic function in determining its shape and vertex?
3. How can you find the maximum height of a projectile given its equation?
4. How does the initial velocity of the projectile affect its maximum height?
5. Can the time to reach maximum height be different if the equation is given in a different form?
6. How does gravity affect the motion of a projectile?
7. What are the steps to convert a quadratic function from standard form to vertex form?
8. How do you determine the total time a projectile is in the air?

**Tip:** Always double-check your calculations, especially when converting fractions to decimals or interpreting values from a formula.

Learn how to find the maximum height of a projectile using the vertex formula of a quadratic function. This problem involves solving for the vertex of the quadratic function -16t^2 + 260t to determine when the projectile reaches its peak height. Suitable for high school students studying quadratic equations and projectile motion.

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