Question

The equation for line g is x+4y=12. Line h is perpendicular to line g and passes through the point (-3,–5). What is the y-intercept of line h?

Answer 1

Alana got a gift card to a movie theater for her birthday. The function f(x)=
–10x+80 represents the balance on Alana's gift card after she sees x movies.
Graph f(x), then answer the question below.

What is the maximum number of movies Alana can see using her gift card?
___ movies

Answer 2

Marcy earns $800 per week for working 40 hours at a sporting goods store. If she works more than 40 hours, she earns overtime pay.
The graph shows Marcy's total weekly earnings for working x hours over 40.

(x-intercept: 800, slope: 3/5, starts at 0 hours and points go by 2s starting by 2)
Which of these statements is true? Select all that apply.

Marcy can earn up to 600$ per week in total weekly savings.
Marcy earns 30$ for ever hour over 40
Marcy can work for as many as 60 hours per week
If Marcy works for 10 hours over 40, she'll earn $300 in overtime pay

Answer 3

To find the \( y \)-intercept of line \( h \), which is perpendicular to line \( g \) and passes through the point \((-3, -5)\), we first need to determine the slope of line \( g \).

The equation of line \( g \) is given as:
\[
x + 4y = 12
\]
We can rewrite this in slope-intercept form \( y = mx + b \) (where \( m \) is the slope and \( b \) is the \( y \)-intercept):
\[
4y = -x + 12
\]
\[
y = -\frac{1}{4}x + 3
\]
From this equation, we see that the slope of line \( g \) is \( -\frac{1}{4} \).

Since line \( h \) is perpendicular to line \( g \), its slope will be the negative reciprocal of the slope of line \( g \). Therefore, the slope of line \( h \) is:
\[
m_h = -\frac{1}{-\frac{1}{4}} = 4
\]

Now we have the slope of line \( h \) and a point on the line \((-3, -5)\). We can use the point-slope form of a line equation to find the equation of line \( h \):
\[
y - y_1 = m_h(x - x_1)
\]
Substituting in the values:
\[
y - (-5) = 4(x - (-3))
\]
\[
y + 5 = 4(x + 3)
\]
\[
y + 5 = 4x + 12
\]
\[
y = 4x + 12 - 5
\]
\[
y = 4x + 7
\]

From this equation, we can see that the \( y \)-intercept (the value of \( b \)) is \( 7 \).

Thus, the \( y \)-intercept of line \( h \) is:
\[
\boxed{7}
\]

Answer 4

To find the maximum number of movies Alana can see using her gift card, we need to find when the balance on her gift card becomes zero. The function \( f(x) = -10x + 80 \) represents the balance on Alana's gift card after she sees \( x \) movies.

To determine how many movies she can see until the balance reaches zero, we will set \( f(x) \) equal to 0:

\[
-10x + 80 = 0
\]

Now, we solve for \( x \):

\[
-10x = -80
\]
\[
x = \frac{-80}{-10}
\]
\[
x = 8
\]

Thus, the maximum number of movies Alana can see using her gift card is **8 movies**.

To confirm the result, we can check the balance after seeing 8 movies:

\[
f(8) = -10(8) + 80 = -80 + 80 = 0
\]

This confirms that after seeing 8 movies, the balance on the gift card is indeed zero.

Thus, the answer is:
\[
\boxed{8}
\] movies.

Answer 5

Let's analyze the provided information step by step.

1. **Understanding Marcy’s earnings**:
- Marcy earns $800 for 40 hours of work.
- For each hour worked over 40, her earnings reflect additional pay.
- According to the graph details, the earnings for hours beyond 40 represent a slope of \(\frac{3}{5}\).
- The slope indicates the change in earnings with respect to hours worked.

2. **Deriving overtime pay**:
- The slope \(\frac{3}{5}\) can be interpreted to mean that for each hour over 40 hours, her total weekly earnings increase by \(\frac{3}{5}(800)\) over 40 hours.
- However, since \(800\) for 40 hours equals \(20\) per hour, it suggests the overtime increase is higher than just \(20\). Thus, to find her actual overtime pay per hour:
\[
\text{Overtime Pay per Hour} = \text{Additional Earnings per Hour} = 0.6 \times 800 / 40 = 30\text{ implies she earns }30\text{ for every overtime hour.}
\]

3. **Analyzing each statement**:
- **Statement 1**: "Marcy can earn up to \$600 per week in total weekly savings."
- This does not seem applicable based on the context provided, which is strictly her earnings regarding work hours; savings aren’t mentioned in relation to her earnings. (Not true)

- **Statement 2**: "Marcy earns \$30 for every hour over 40."
- This is true because we derived that her overtime pay is indeed \$30 per hour for every hour over 40. (True)

- **Statement 3**: "Marcy can work for as many as 60 hours per week."
- This is generally true unless specified otherwise in terms of maximum total hours the store allows. In the absence of such limitations, working up to 60 hours is reasonable. (True)

- **Statement 4**: "If Marcy works for 10 hours over 40, she'll earn \$300 in overtime pay."
- If Marcy works 10 hours over 40, her overtime pay calculation would be:
\[
10 \text{ hours} \times 30 = 300
\]
- This statement is true. (True)

So, the correct true statements are:

- **Marcy earns \$30 for every hour over 40.**
- **Marcy can work for as many as 60 hours per week.**
- **If Marcy works for 10 hours over 40, she'll earn \$300 in overtime pay.**

Thus the selected true statements are:
- Marcy earns \$30 for every hour over 40.
- Marcy can work for as many as 60 hours per week.
- If Marcy works for 10 hours over 40, she'll earn \$300 in overtime pay.