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the equation is y = 1.2x + 30

Drag and drop to fill in the blanks.

Consider the data shown on the graph.

• The y-intercept represents a temperature of ( Response area)°F when there are (Response area) cricket chirps.
• The slope represents an additional (Response area) chirps for each additional (Response area)°F increase in temperature.
• If the number of chirps increases from 10 to 30, the temperature has increased by (Response area)°F.
• If the temperature increases from 30°F to 60°F, the number of chirps has increased by (Response area) chirps.

responses
24
1
30
1.2
25
0

11 answers

Here's how to fill in the blanks based on the given equation \( y = 1.2x + 30 \):

  • The y-intercept represents a temperature of 30°F when there are 0 cricket chirps.
  • The slope represents an additional 1.2 chirps for each additional 1°F increase in temperature.
  • If the number of chirps increases from 10 to 30, the temperature has increased by 24°F.
  • If the temperature increases from 30°F to 60°F, the number of chirps has increased by 30 chirps.

Now, let's place the responses in the correct places:

• The y-intercept represents a temperature of 30°F when there are 0 cricket chirps.
• The slope represents an additional 1.2 chirps for each additional 1°F increase in temperature.
• If the number of chirps increases from 10 to 30, the temperature has increased by 24°F.
• If the temperature increases from 30°F to 60°F, the number of chirps has increased by 30 chirps.

(Note: "0" isn't provided among the response options, so you may want to check if that area allows "0" as an acceptable answer or if it's only looking for numbers given in the options.)

you can only use 30 once you forgot about 25

Thank you for the clarification! Based on the constraints, here is the correct way to fill in the blanks:

  • The y-intercept represents a temperature of 30°F when there are 0 cricket chirps.
  • The slope represents an additional 1.2 chirps for each additional 1°F increase in temperature.
  • If the number of chirps increases from 10 to 30, the temperature has increased by 24°F.
  • If the temperature increases from 30°F to 60°F, the number of chirps has increased by 30 chirps.

However, since you mentioned that we can only use the number 30 once and need to incorporate 25, here’s how we can do that:

  1. y-intercept: The y-intercept is 30°F when there are 0 chirps.
  2. slope: The slope represents an additional 1.2 chirps for each additional 1°F increase in temperature.
  3. An increase from 10 to 30 chirps corresponds to an increase of 24°F.
  4. An increase from 30°F to 60°F corresponds to an increase of 30 chirps.

To bring in 25 in a meaningful way, you could clarify that the average number of chirps when the temperature is around that range might be relevant, but we generally still need the direct answers.

If we assign 1 chirp as a temperature or another context that fits better:

  1. The y-intercept represents a temperature of 30°F when there are 0 cricket chirps.
  2. The slope represents an additional 1.2 chirps for each additional 1°F increase in temperature.
  3. If the number of chirps increases from 10 to 30, the temperature has increased by 24°F.
  4. If the temperature increases from 30°F to 60°F, you could discuss chirps could rise significantly but not use 25 directly.

However, for strict filling using given values and aiming at the best combination while respecting your constraints and mentioning 25 where needed, you can adjust queries around that.

Please confirm where you need the 25 value to make it fit the problem, or if it's allowed in a non-direct way!

the equation is y = 0.70x + 200

what does the rate in change represent in context of the data?

responses
the price of airfair increases by $0.30 for each mile traveled.
the price of airfair decreases by $0.30 for each mile traveled.
the price of airfare increases by $0.70 for each mile traveled.
the price of airfare decreases by $0.70 for each mile traveled.

In the equation \( y = 0.70x + 200 \), the slope (rate of change) is \( 0.70 \). This value represents how much the price of airfare changes for each unit increase in \( x \).

Here, \( x \) likely represents the number of miles traveled and \( y \) represents the total price of airfare.

Thus, the correct interpretation of the rate of change is:

The price of airfare increases by $0.70 for each mile traveled.

So you should select that response.

number of mint candies 1 1 1 3 3 3
height of spurt (centimeters) 40 35 30 110 105 90

number of mint candies 5 5 5 10 10 10
height of spurt (centimeters) 170 160 180 400 390 420
Gary saw a video on the internet in which dropping mint candies into bottles of soda pop caused the soda pop to spurt immediately from the bottle. He wondered if the height of the spurt was linearly related to the number of mint candies used. He collected data using 1, 3, 5, and 10 mint candies dropped into two-liter bottles of soda pop. The height of the spurt was measured in centimeters. Each quantity of mint candies was tried three times. The data is shown in the table.

Find the mean height of the spurt for 3 mint candies and the mean height of the spurt for 10 candies. Then, use these two data points to estimate the line of fit for the data. What is the BEST interpretation of the y-intercept in context of this problem?
Responses
A The y-intercept is (0, −8.1). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −8.1 cm. This makes sense within the context of the problem.The y-intercept is (0, −8.1). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −8.1 cm. This makes sense within the context of the problem.
B The y-intercept is (0, −8.1). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −8.1 cm. This does not make sense within the context of the problem.The y-intercept is (0, −8.1). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −8.1 cm. This does not make sense within the context of the problem.
C The y-intercept is (0, −27.6). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −27.6 cm. This makes sense within the context of the problem.The y-intercept is (0, −27.6). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −27.6 cm. This makes sense within the context of the problem.
D The y-intercept is (0, −27.6). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −27.6 cm. This does not make sense within the context of the problem.The y-intercept is (0, −27.6). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −27.6 cm. This does not make sense within the context of the problem.
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To analyze the data provided, let’s first calculate the mean height of the spurt for 3 mint candies and for 10 mint candies.

Data Points for 3 Mint Candies:

  • Heights: 40 cm, 35 cm, 30 cm
  • Mean height = (40 + 35 + 30) / 3 = 105 / 3 = 35 cm

Data Points for 10 Mint Candies:

  • Heights: 400 cm, 390 cm, 420 cm
  • Mean height = (400 + 390 + 420) / 3 = 1210 / 3 = 403.33 cm (approximately)

Now we have the following two data points for the line of best fit:

  • (3, 35)
  • (10, 403.33)

Next, we can find the slope (\(m\)) of the line of best fit using the two points (3, 35) and (10, 403.33):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{403.33 - 35}{10 - 3} = \frac{368.33}{7} \approx 52.61 \]

Now using the point-slope form of the equation of a line, we can use one of the points to find the y-intercept (\(b\)). We can use the point (3, 35):

\[ y - 35 = 52.61(x - 3) \]

Calculating for when \(x = 0\) (to find the y-intercept): \[ y - 35 = 52.61(0 - 3) \] \[ y - 35 = -157.83 \] \[ y = 35 - 157.83 = -122.83 \]

So, the y-intercept is approximately \((0, -122.83)\).

Interpretation of the y-intercept

The y-intercept of the regression line represents the predicted height of the spurt if no mint candies are dropped into the bottle.

Since the y-intercept yields a negative height, it suggests that if no mint candies are added, the height of the spurt would be \(−122.83\) cm, which doesn't make sense in the context of the problem because height cannot be negative.

Best Interpretation

Given the options regarding the y-intercept, based on this reasoning, the appropriate response would be:

B) The y-intercept is (0, −8.1). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is −8.1 cm. This does not make sense within the context of the problem.

(Note: If the calculations around y-intercept yield a different value from what's presented in the options, please revisit the linear equation calculation or specifics to ensure alignment.)

A botanist collected data on the growth of sunflowers to maturity. She modeled the equation y = x + 1/2 for the line of best fit. The independent variable, x, is time in weeks and the dependent variable, y, is height in feet. What does the slope mean in this context?
Responses
A The sunflower's starting height is 1 foot.The sunflower's starting height is 1 foot.
B The sunflower's starting height is 1/2 foot.
C The sunflower's height increases by an additional 1 foot each week.The sunflower's height increases by an additional 1 foot each week.
D The sunflower's height increases by an additional 1/2 foot each week.

In the equation \( y = x + \frac{1}{2} \):

  • \( y \) represents the height of the sunflower in feet (dependent variable).
  • \( x \) represents time in weeks (independent variable).
  • The slope of the line in this equation is 1 (the coefficient of \( x \)).

Interpretation of the Slope

The slope indicates the rate of change of the dependent variable (height) with respect to the independent variable (time). In this case, a slope of 1 means that for each additional week (\( x \)), the height of the sunflower (\( y \)) increases by 1 foot.

Therefore, the correct interpretation is:

C) The sunflower's height increases by an additional 1 foot each week.

length (minutes) 1.5 2 3 4.5
amount of water (gallons) 3,700 4,100 6,450 8,400

Old Faithful is a geyser in Yellowstone National Park. The table shows the approximate length of an eruption (in minutes) and the amount of water (in gallons) in that eruption.

What is the BEST interpretation of the slope in context of this problem?
Responses
A Each additional minute in eruption length results in a prediction of an additional 1,350 gallons of water produced.Each additional minute in eruption length results in a prediction of an additional 1,350 gallons of water produced.
B Each additional minute in eruption length results in a prediction of an additional 1,567 gallons of water produced.Each additional minute in eruption length results in a prediction of an additional 1,567 gallons of water produced.
C That the minimum amount of water produced for each additional minute in eruption length is 1,867 gallons.That the minimum amount of water produced for each additional minute in eruption length is 1,867 gallons.
D That the maximum amount of water produced for each additional minute in eruption length is 2,467 gallons.

To determine the best interpretation of the slope in the context of the data provided, we first need to calculate the slope of the line of best fit based on the given data points.

We have the following data points:

  • (1.5, 3,700)
  • (2, 4,100)
  • (3, 6,450)
  • (4.5, 8,400)

Step 1: Calculate the Slope

To find the slope of the line of best fit, we can use any two points, but it's typically best to use points that are further apart for a more accurate calculation. Let's calculate the slope \( m \) using the first and last data points:

  1. The change in gallons of water (\( \Delta y \)) between the points (1.5, 3,700) and (4.5, 8,400): \[ \Delta y = 8,400 - 3,700 = 4,700 \text{ gallons} \]

  2. The change in time in minutes (\( \Delta x \)): \[ \Delta x = 4.5 - 1.5 = 3 \text{ minutes} \]

  3. Now, calculate the slope: \[ m = \frac{\Delta y}{\Delta x} = \frac{4,700}{3} \approx 1,566.67 \text{ gallons per minute} \]

Step 2: Interpret the Slope

The slope of approximately 1,567 gallons per minute means that for each additional minute of eruption length, the amount of water produced increases by about 1,567 gallons.

Conclusion

Thus, the best interpretation of the slope in the context of this problem is:

B) Each additional minute in eruption length results in a prediction of an additional 1,567 gallons of water produced.