A brick is dropped into a lake, 450 m deep, and its depth d (in m) from the top of the lake is recorded at 1-minute intervals. Based on these observations, the height and the time can be modelled using the following function:
d left parenthesis t right parenthesis equals 4.90 t squared minus 87.27 t minus 61

b. Predict the time when the brick hits the bottom of the lake.

To find the time when the brick hits the bottom of the lake, we need to determine when the depth \( d(t) \) is equal to the depth of the lake, which is 450 meters.

The function given is:
\[
d(t) = 4.90t^2 - 87.27t - 61
\]

To find when the brick reaches the bottom, set \( d(t) = 450 \) and solve for \( t \):

\[
450 = 4.90t^2 - 87.27t - 61
\]

Rearrange the equation to set it equal to zero:

\[
4.90t^2 - 87.27t - 511 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = 4.90 \)
- \( b = -87.27 \)
- \( c = -511 \)

Substitute these values into the formula:

\[
t = \frac{-(-87.27) \pm \sqrt{(-87.27)^2 - 4(4.90)(-511)}}{2(4.90)}
\]

Now calculate the discriminant and the time \( t \).The solutions for the time \( t \) are approximately:

- \( t_1 = 22.45 \) minutes
- \( t_2 = -4.64 \) minutes

Since time cannot be negative, the only physically meaningful solution is \( t_1 \).

Thus, the brick will hit the bottom of the lake approximately **22.45 minutes** after being dropped.

Would you like further details or have any questions?

Here are five related questions:
1. How would the time change if the lake were deeper?
2. What would happen if the initial velocity were different?
3. How can the time be calculated if the brick is thrown upward first?
4. How would air resistance affect the time it takes to reach the bottom?
5. How does the quadratic nature of the equation relate to the motion of the brick?

**Tip:** When dealing with quadratic equations, always consider the physical context to determine which solutions are meaningful.

Explore how to predict the time when a brick hits the bottom of a lake using a quadratic equation. Step-by-step solution provided with detailed calculations. Suitable for high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation