A man using a telescope is watching the path of the moon. A vee line blocks the view of the moon for
part of the night. Viewed from his telescope, the path of the moon can be represented by the function
. The tree line can be represented by the function , where is the
vertical position, in pixels, of his viewing window, and is the time in hours since 7 P.M. To the nearest
hour, how long is the moon visible to the man through his telescope?

Ask by Huang Santiago. in the United States

Mar 30,2025

Question

A man using a telescope is watching the path of the moon. A vee line blocks the view of the moon for
part of the night. Viewed from his telescope, the path of the moon can be represented by the function
. The tree line can be represented by the function , where is the
vertical position, in pixels, of his viewing window, and is the time in hours since 7 P.M. To the nearest
hour, how long is the moon visible to the man through his telescope?

Ask by Huang Santiago. in the United States

Mar 30,2025

UpStudy AI · Accepted Answer

We are given the moon’s path as  
$$
y = -3x^2 + 33x - 30,
$$  
and the tree line as  
$$
y = 0.6x + 10,
$$  
where $$ x $$ is the number of hours after 7 P.M. The moon is visible when its altitude as seen through the telescope is above the tree line, so we must find when

$$
-3x^2 + 33x - 30 = 0.6x + 10.
$$

**Step 1. Set up the equation**

Subtract $$ 0.6x + 10 $$ from both sides to obtain

$$
-3x^2 + 33x - 30 - 0.6x - 10 = 0,
$$

which simplifies to

$$
-3x^2 + 32.4x - 40 = 0.
$$

It is easier to work with a positive quadratic coefficient. Multiply the entire equation by $$-1$$:

$$
3x^2 - 32.4x + 40 = 0.
$$

**Step 2. Solve the quadratic equation**

Use the quadratic formula

$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},
$$

with $$a=3$$, $$b=-32.4$$, and $$c=40$$. Substituting these values,

$$
x=\frac{-(-32.4)\pm\sqrt{(-32.4)^2-4\cdot3\cdot40}}{2\cdot3} = \frac{32.4\pm\sqrt{32.4^2-480}}{6}.
$$

Calculate the discriminant:

$$
32.4^2 = 1049.76,
$$

so

$$
1049.76-480 = 569.76.
$$

Taking the square root:

$$
\sqrt{569.76}\approx 23.87.
$$

Thus, the solutions are

$$
x = \frac{32.4 \pm 23.87}{6}.
$$

**Step 3. Find the two intersection points**

Compute the two solutions:

- For the negative sign:
  $$
  x = \frac{32.4 - 23.87}{6} \approx \frac{8.53}{6} \approx 1.42.
  $$
- For the positive sign:
  $$
  x = \frac{32.4 + 23.87}{6} \approx \frac{56.27}{6} \approx 9.38.
  $$

**Step 4. Determine the duration the moon is above the tree line**

The moon is visible from when $$ x \approx 1.42 $$ hours after 7 P.M. until $$ x \approx 9.38 $$ hours after 7 P.M. The duration of visibility is

$$
9.38 - 1.42 \approx 7.96 \text{ hours}.
$$

Rounded to the nearest hour, the moon is visible for

$$
\approx 8 \text{ hours}.
$$
The moon is visible for approximately 8 hours through the telescope.

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