https://file.aiquickdraw.com/mathbot/file/1727465006673lwn0pd1q.jpg

Let's go through the problem step by step based on the image you've provided.

The general position equation is given by:
\[
s = -16t^2 + v_0t + s_0
\]
where:
- \( s \) is the height of the object at time \( t \) (in feet),
- \( v_0 \) is the initial velocity (in feet per second),
- \( s_0 \) is the initial height (in feet),
- \( t \) is the time in seconds.

### (a) Write the mathematical model for the height of the coin.

Since the coin is dropped from a building, the initial velocity \( v_0 \) is 0, and the initial height \( s_0 \) is 940 feet (the height of the building). So, the equation becomes:
\[
s = -16t^2 + 940
\]
This is the mathematical model for the height of the coin as a function of time.

### (b) Find the height of the coin after 6 seconds.

We can substitute \( t = 6 \) into the equation:
\[
s = -16(6)^2 + 940
\]
Let's calculate it.

\[
s = -16(36) + 940 = -576 + 940 = 364 \text{ feet}
\]
So, the height of the coin after 6 seconds is **364 feet**.

### (c) Find how long it takes the coin to strike the ground.

To find when the coin strikes the ground, set \( s = 0 \) (since the coin reaches the ground when the height is 0):
\[
0 = -16t^2 + 940
\]
Rearranging this equation:
\[
16t^2 = 940
\]
\[
t^2 = \frac{940}{16} = 58.75
\]
Now, take the square root:
\[
t = \sqrt{58.75} \approx 7.66 \text{ seconds}
\]
So, it takes approximately **7.66 seconds** for the coin to strike the ground.

---

Would you like further details on any part of this explanation? Here are 5 related questions to expand your understanding:

1. How does the initial velocity affect the motion of the object?
2. What would the equation look like if the object were thrown upwards with an initial velocity?
3. How does the constant \(-16\) in the equation relate to gravity?
4. What other real-world scenarios could use this type of position equation?
5. How would the time to hit the ground change if the object were dropped from a different height?

**Tip:** When solving quadratic equations like in part (c), you can always use the quadratic formula if the equation doesn’t simplify easily.

You drop a coin from the top of a building. The building has a height of 940 feet. Use the position equation to write a mathematical model for the height of the coin. (a) Find the mathematical model for the height of the coin. (b) Find the height of the coin after 6 seconds. (c) How long does it take the coin to strike the ground?

Explore how to solve a real-world free fall problem by using quadratic equations. Learn to calculate the height of a dropped coin after a specific time and determine the time taken for it to hit the ground from a 940-foot building. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation