Question
A diver jumps off a platform at an initial upward velocity of 20 feet per second into the air above a pool. The height of the diver above the water after jumping can be represented by the function: h(t)=−16t2+20t Use desmos to graph the function. Identify the x- intercept and interpret its
Answers
GPT-4o mini
To analyze the function \( h(t) = -16t^2 + 20t \), we're interested in identifying the x-intercepts, which represent the times when the diver is at the height of zero (i.e., when the diver hits the water).
### Step 1: Find the x-intercepts
To find the x-intercepts, we need to set \( h(t) = 0 \):
\[
-16t^2 + 20t = 0
\]
Factoring out \( t \):
\[
t(-16t + 20) = 0
\]
This gives us two factors to solve:
1. \( t = 0 \)
2. \( -16t + 20 = 0 \)
For the second factor, we can solve for \( t \):
\[
-16t + 20 = 0 \implies -16t = -20 \implies t = \frac{20}{16} = \frac{5}{4} = 1.25 \text{ seconds}
\]
### Step 2: Interpret the x-intercepts
The x-intercepts of the graph of the function \( h(t) \) are at:
- **\( t = 0 \)**: This intercept indicates the time when the diver just jumped off the platform.
- **\( t = 1.25 \)**: This intercept indicates the time when the diver hits the water after 1.25 seconds of being in the air.
### Summary of interpretation:
- The diver starts at a height of 0 feet (when \( t = 0 \)).
- The diver will reach the water again after 1.25 seconds, as shown by the second x-intercept.
By graphing the function \( h(t) = -16t^2 + 20t \) in Desmos (or any graphing tool), you will visually confirm that these are indeed the x-intercepts of the parabola representing the diver's height over time. The graph will show a downward-opening parabola intersecting the x-axis at these points.
### Step 1: Find the x-intercepts
To find the x-intercepts, we need to set \( h(t) = 0 \):
\[
-16t^2 + 20t = 0
\]
Factoring out \( t \):
\[
t(-16t + 20) = 0
\]
This gives us two factors to solve:
1. \( t = 0 \)
2. \( -16t + 20 = 0 \)
For the second factor, we can solve for \( t \):
\[
-16t + 20 = 0 \implies -16t = -20 \implies t = \frac{20}{16} = \frac{5}{4} = 1.25 \text{ seconds}
\]
### Step 2: Interpret the x-intercepts
The x-intercepts of the graph of the function \( h(t) \) are at:
- **\( t = 0 \)**: This intercept indicates the time when the diver just jumped off the platform.
- **\( t = 1.25 \)**: This intercept indicates the time when the diver hits the water after 1.25 seconds of being in the air.
### Summary of interpretation:
- The diver starts at a height of 0 feet (when \( t = 0 \)).
- The diver will reach the water again after 1.25 seconds, as shown by the second x-intercept.
By graphing the function \( h(t) = -16t^2 + 20t \) in Desmos (or any graphing tool), you will visually confirm that these are indeed the x-intercepts of the parabola representing the diver's height over time. The graph will show a downward-opening parabola intersecting the x-axis at these points.