To analyze this problem accurately, we'll identify the components essential for modeling the system of equations represented by the motion of the worker in the bucket lift and the measuring tape being thrown.
First, let's consider the worker being raised in a bucket lift.
The worker is being raised at a constant speed of [tex]\( 3 \, \text{ft/s} \)[/tex]. When the worker's arms are 10 feet off the ground, this serves as the initial height.
For an object moving at a constant speed, the formula for height [tex]\( h \)[/tex] as a function of time [tex]\( t \)[/tex] is:
[tex]\[ h = \text{initial height} + (\text{speed} \times \text{time}) \][/tex]
Plugging in the given numbers:
[tex]\[ h = 10 + 3t \][/tex]
Thus, the height model for the worker is:
[tex]\[ h = 3t + 10 \][/tex]
Next, let's consider the measuring tape being thrown.
The measuring tape is thrown with an initial vertical velocity of [tex]\( 15 \, \text{ft/s} \)[/tex] from a height of [tex]\( 6 \, \text{ft} \)[/tex]. This motion follows the projectile motion formula given by:
[tex]\[ h = -16t^2 + vt + h_0 \][/tex]
Where:
- [tex]\( v \)[/tex] is the initial vertical velocity (15 ft/s),
- [tex]\( h_0 \)[/tex] is the initial height (6 ft),
- [tex]\( t \)[/tex] is the time in seconds,
- [tex]\( h \)[/tex] is the height in feet.
Substitute the given values into the formula:
[tex]\[ h = -16t^2 + 15t + 6 \][/tex]
Thus, the height model for the measuring tape is:
[tex]\[ h = -16t^2 + 15t + 6 \][/tex]
Considering both models together, we have the system of equations:
[tex]\[ h = 3t + 10 \][/tex]
[tex]\[ h = -16t^2 + 15t + 6 \][/tex]
Comparing with the options given in the question:
1. [tex]\( h = 3t + 10 \)[/tex] and [tex]\( h = -16t^2 + 15t + 6 \)[/tex]
2. [tex]\( 10t + 3 \)[/tex] and [tex]\( h = -16t^2 + 6t + 15 \)[/tex]
3. [tex]\( -16t^2 + 3t + 10 \)[/tex] and [tex]\( h = -16t^2 + 15t + 6 \)[/tex]
4. [tex]\( -16t^2 + 10t + 3 \)[/tex] and [tex]\( h = -16t^2 + 6t + 15 \)[/tex]
We see that the correct answer is:
[tex]\[ h = 3t + 10 \][/tex] and [tex]\[ h = -16t^2 + 15t + 6 \][/tex]
So, the correct system of equations modeling this situation is:
[tex]\[ h = 3t + 10 \][/tex] and [tex]\( h = -16t^2 + 15t + 6 \)[/tex]
