To solve this problem, we need to determine the horizontal distance (or range) where the blood drop landed from the point of impact. We are given:
- The angle of impact, [tex]\(\theta = 56^\circ\)[/tex]
- The height from which the drop fell, [tex]\(h = 5\)[/tex] feet
We will assume the blood droplet follows a straight line trajectory when it comes off from the point of impact. In this case, we can use basic trigonometry to calculate the horizontal distance.
1. Understanding the problem: The blood droplet falls from a height with an angle to the ground. We can think of this scenario as forming a right-angled triangle where:
- The height ([tex]\(h\)[/tex]) is the vertical leg.
- The horizontal distance ([tex]\(d\)[/tex]) is the horizontal leg.
- The angle of impact ([tex]\(\theta\)[/tex]) is between the horizontal leg and the hypotenuse.
2. Trigonometric relationship: We use the tangent function, which is defined as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here:
- Opposite side = height ([tex]\(h\)[/tex])
- Adjacent side = horizontal distance ([tex]\(d\)[/tex])
So we can state:
[tex]\[
\tan(56^\circ) = \frac{5}{d}
\][/tex]
3. Solve for [tex]\(d\)[/tex]: Rearrange the equation to solve for the horizontal distance:
[tex]\[
d = \frac{5}{\tan(56^\circ)}
\][/tex]
4. Calculate [tex]\(\tan(56^\circ)\)[/tex]:
Using a calculator, we find:
[tex]\[
\tan(56^\circ) \approx 1.4826
\][/tex]
5. Compute the horizontal distance:
[tex]\[
d = \frac{5}{1.4826} \approx 3.372 \text{ feet}
\][/tex]
Therefore, the blood drop landed approximately 3.372 feet away from the point of impact.
