Sure, let's carefully work through the problem step-by-step to determine how high Avi will reach after compressing the trampoline.
### Step-by-Step Solution:
1. Given Data:
- Mass of Avi: [tex]\( m = 40 \, \text{kg} \)[/tex]
- Spring constant of the trampoline: [tex]\( k = 176{,}400 \, \frac{\text{N}}{\text{m}} \)[/tex]
- Compression of the trampoline: [tex]\( x = 0.20 \, \text{m} \)[/tex] (Note: 20 cm is converted to meters)
2. Calculate the Potential Energy Stored in the Spring:
The potential energy stored in a compressed spring is given by the formula:
[tex]\[
PE_\text{spring} = \frac{1}{2} k x^2
\][/tex]
Plugging in the given values:
[tex]\[
PE_\text{spring} = \frac{1}{2} \times 176{,}400 \, \frac{\text{N}}{\text{m}} \times (0.20 \, \text{m})^2
\][/tex]
[tex]\[
PE_\text{spring} = \frac{1}{2} \times 176{,}400 \times 0.04
\][/tex]
[tex]\[
PE_\text{spring} = 3{,}528 \, \text{J} \quad (\text{Joules})
\][/tex]
3. Convert the Potential Energy into Height:
The potential energy stored in the spring will be converted into gravitational potential energy at the peak of the jump. Gravitational potential energy is given by:
[tex]\[
PE_\text{gravity} = mgh
\][/tex]
We set the spring’s potential energy equal to the gravitational potential energy to find the height [tex]\( h \)[/tex]:
[tex]\[
PE_\text{spring} = PE_\text{gravity}
\][/tex]
[tex]\[
3{,}528 = 40 \, \text{kg} \times 9.81 \, \frac{\text{m}}{\text{s}^2} \times h
\][/tex]
Rearrange the equation to solve for [tex]\( h \)[/tex]:
[tex]\[
h = \frac{3{,}528}{40 \times 9.81}
\][/tex]
4. Perform the Calculation:
[tex]\[
h = \frac{3{,}528}{392.4}
\][/tex]
[tex]\[
h \approx 8.99 \, \text{meters}
\][/tex]
### Conclusion:
Avi should reach a height of approximately 8.99 meters from the point of maximum trampoline compression.
Thus, the answer is:
[tex]\[
\boxed{8.99 \, \text{meters}}
\][/tex]
