are 10 ft off the ground, her coworker throws a measuring tape toward her. The measuring tape is thrown from a height of 6 ft with an initial vertical velocity of $$ 15 \mathrm{ft} / \mathrm{s} $$. Projectile motion formula: $$ h=-16 t^{2}+v t+h_{0} $$ $$ t= $$ time, in seconds, since the measuring tape was thrown $$ h= $$ height, in feet, above the ground $$ h=3 t+10 $$ and $$ h=-16 t^{2}+15 t+6 $$ $$ 10 t+3 $$ and $$ h=-16 t^{2}+6 t+15 $$ $$ -16 t^{2}+3 t+10 $$ and $$ h=-16 t^{2}+15 t+6 $$ $$ -16 t^{2}+10 t+3 $$ and $$ h=-16 t^{2}+6 t+15 $$ COMPLETE The system has no real solutions. What does this indicate about the worker's arms and the measuring tape? They will never be at the same height at the same time. They will be at the same height exactly once. They will be at the same height at two different times. DONE

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UpStudy AI · Accepted Answer

To determine the relationship between the heights of the worker and the measuring tape, we need to analyze the equations given for their heights:

1. The height of the worker is given by:
   $$
   h_w = 3t + 10
   $$
   where $$ h_w $$ is the height of the worker in feet.

2. The height of the measuring tape is given by:
   $$
   h_t = -16t^2 + 15t + 6
   $$
   where $$ h_t $$ is the height of the measuring tape in feet.

To find out if and when they are at the same height, we need to set these two equations equal to each other:
$$
3t + 10 = -16t^2 + 15t + 6
$$

Now, let's rearrange this equation to form a standard quadratic equation:
$$
16t^2 + 3t - 15t + 10 - 6 = 0
$$
$$
16t^2 - 12t + 4 = 0
$$

Next, we can simplify this equation:
$$
4t^2 - 3t + 1 = 0
$$

Now, we will use the discriminant $$ D $$ to determine the nature of the solutions:
$$
D = b^2 - 4ac
$$
where $$ a = 4 $$, $$ b = -3 $$, and $$ c = 1 $$.

Calculating the discriminant:
$$
D = (-3)^2 - 4(4)(1) = 9 - 16 = -7
$$

Since the discriminant $$ D $$ is negative, this indicates that the quadratic equation has no real solutions.

### Conclusion:
The fact that there are no real solutions means that the worker's arms and the measuring tape will never be at the same height at the same time. Thus, the correct interpretation is:

**They will never be at the same height at the same time.**
They will never be at the same height at the same time.

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