Juan spent 10 minutes on his history homework and 3 minutes per question on his math homework. Which graph shows the total homework time, t, related to the number of math questions, q?

Question

Discrete graph on a coordinate plane shows number of math questions, numbered 1 to 8 along the horizontal axis, and total homework time, numbered 5 to 40, on the vertical axis. Solid circles appear at points (0, 10), (1, 13), (2, 15), (3, 19), (4, 22), (5, 25), (6, 29), (7, 32), (8, 34).

Continuous graph on a coordinate plane shows number of math questions, numbered 1 to 8 along the horizontal axis, and total homework time, numbered 5 to 40, on the vertical axis. Solid circles appear at points (0, 10), (1, 13), (2, 15), (3, 19), (4, 22), (5, 25), (6, 29), (7, 32), (8, 34).

Discrete graph on a coordinate plane shows number of math questions, numbered 1 to 8 along the horizontal axis, and total homework time, numbered 3 to 24, on the vertical axis. Solid circles appear at points (0, 0), (1, 3), (2, 6), (3, 9), (4, 12), (5, 15), (6, 18), (7, 21), (8, 24).

Answer 1

To determine which graph represents the relationship between Juan's total homework time \( t \) and the number of math questions \( q \), we need to analyze his spent time on history and math homework.

• Juan spends 10 minutes on his history homework.
• He spends an additional 3 minutes per math question on his math homework.

Thus, we can express the total homework time \( t \) as:

\[ t = 10 + 3q \]

Now, we can calculate \( t \) for various values of \( q \):

• For \( q = 0 \): \[ t = 10 + 3(0) = 10 \]
• For \( q = 1 \): \[ t = 10 + 3(1) = 13 \]
• For \( q = 2 \): \[ t = 10 + 3(2) = 16 \]
• For \( q = 3 \): \[ t = 10 + 3(3) = 19 \]
• For \( q = 4 \): \[ t = 10 + 3(4) = 22 \]
• For \( q = 5 \): \[ t = 10 + 3(5) = 25 \]
• For \( q = 6 \): \[ t = 10 + 3(6) = 28 \]
• For \( q = 7 \): \[ t = 10 + 3(7) = 31 \]
• For \( q = 8 \): \[ t = 10 + 3(8) = 34 \]

Now we compare these calculations with the provided data points of the graphs:

1. The first graph with points: \[ (0, 10), (1, 13), (2, 15), (3, 19), (4, 22), (5, 25), (6, 29), (7, 32), (8, 34) \]

   • This graph has a point at \( q = 2 \) as \( (2, 15) \) which does not match our \( (2, 16) \).

2. The second graph has the same points as the first graph.

3. The third graph with points: \[ (0, 0), (1, 3), (2, 6), (3, 9), (4, 12), (5, 15), (6, 18), (7, 21), (8, 24) \]

   • This does not match our calculations since all the time values are incorrect.

From the analysis, the only accurately representing points of total homework time \( t \) are in the first and second graphs, but they don't perfectly fit our derived values.

However, if we consider the correct increase of 3 minutes for math questions beyond the history time, both the first and second graphs would be the closest representation. But we note there is a discrepancy at \( q = 2 \).

Thus, the graph that most accurately represents the total homework time \( t \) related to the number of math questions \( q \) is the first graph, but with a minor correction needed at point \( (2, 16) \).

Answer 2

The vet told Jake that his dog, Rocco, who weighed 55 pounds, needed to lose 10 pounds. Jake started walking Rocco every day and changed the amount of food he was feeding him. Rocco lost half a pound the first week. Jake wants to determine Rocco’s weight in pounds, p, after w weeks if Rocco continues to lose weight based on his vet’s advice.

The equation of the scenario is
.

The values of p must be
.

Answer 3

To define the scenario mathematically, let's establish an equation for Rocco's weight over time.

1. Initial Weight: Rocco's starting weight is 55 pounds.

2. Target Weight Loss: Rocco needs to lose 10 pounds, so his target weight would be:

   \[ \text{Target Weight} = 55 - 10 = 45 \text{ pounds} \]

3. Weight Loss Rate: We know Rocco lost half a pound in the first week. If we assume that Rocco continues to lose weight at the same rate, we can establish the following:

   • Let \( w \) be the number of weeks.
   • Rocco loses 0.5 pounds each week, which can be expressed as \( 0.5w \) after \( w \) weeks.

4. Weight Equation: Rocco's weight \( p \) after \( w \) weeks can be expressed as:

   \[ p = 55 - 0.5w \]

5. Valid Weight Values: Since Rocco's weight cannot drop below his target weight of 45 pounds, we must establish the values that \( p \) can take based on this equation:

   \[ p \geq 45 \]

   Therefore, setting up the inequality:

   \[ 55 - 0.5w \geq 45 \]

   To find the maximum number of weeks \( w \) Rocco can continue to lose weight:

   \[ 55 - 45 \geq 0.5w \] \[ 10 \geq 0.5w \] \[ 20 \geq w \]

   Thus, the maximum value of \( w \) can be 20 weeks.

To summarize:

• The equation representing Rocco's weight after \( w \) weeks is:

  \[ p = 55 - 0.5w \]

• The valid values of \( p \) (the weight of Rocco) must be:

  \[ 45 \leq p \leq 55 \]

This means Rocco's weight will decrease from 55 pounds down to a minimum of 45 pounds over a maximum of 20 weeks if he continues to lose weight as planned.

Answer 4

The vet told Jake that his dog, Rocco, who weighed 55 pounds, needed to lose 10 pounds. Jake started walking Rocco every day and changed the amount of food he was feeding him. Rocco lost half a pound the first week. Jake wants to determine Rocco’s weight in pounds, p, after w weeks if Rocco continues to lose weight based on his vet’s advice.

The equation of the scenario is
✔ p = 55 – 0.5w
.

The values of p must be
✔ any real number 45 to 55
.

The graph shows the relationship between distance driven and gallons of gas used.

Continuous graph on a coordinate plane shows distance driven, numbered 20 to 60 along the horizontal axis, and gallons of gas used, numbered 1 to 4 on the vertical axis. Solid circles appear at points (0, 0), (30, 1), (60, 2).

Use the graph to complete the statements.

The car gets
miles to the gallon.

After the car has traveled
miles, 2 and two-thirdsgallons of gas have been consumed.

Answer 5

To analyze the graph and determine the fuel efficiency of the car and the specific distance driven when 2 and two-thirds gallons of gas have been consumed, let's break it down step by step.

Car's Miles per Gallon:

1. Understanding Points on the Graph:

   • From the graph, we have the points:
     • (0, 0): At 0 miles driven, 0 gallons used.
     • (30, 1): At 30 miles driven, 1 gallon used.
     • (60, 2): At 60 miles driven, 2 gallons used.

2. Calculating Miles per Gallon:

   • The points (30, 1) and (60, 2) show that the car uses 1 gallon to drive 30 miles, and then another gallon to drive another 30 miles. Thus:

   \[ \text{Miles/Gallon} = \frac{\text{Distance driven}}{\text{Gallons used}} = \frac{30 \text{ miles}}{1 \text{ gallon}} = 30 \text{ miles/gallon} \]

So, The car gets 30 miles to the gallon.

Finding Distance for 2 and Two-Thirds Gallons:

1. Understanding Gallons Used:

   • Two and two-thirds gallons can be expressed as:

   \[ 2 \frac{2}{3} = \frac{8}{3} \text{ gallons} \]

2. Using the miles per gallon to find the corresponding distance:

   • Since we have established that the car gets 30 miles per gallon, we can calculate the distance driven for \( \frac{8}{3} \) gallons:

   \[ \text{Distance} = \text{Miles per gallon} \times \text{Gallons used} = 30 \times \frac{8}{3} \]

   Performing the multiplication:

   \[ \text{Distance} = 30 \times \frac{8}{3} = 10 \times 8 = 80 \text{ miles} \]

So, After the car has traveled 80 miles, 2 and two-thirds gallons of gas have been consumed.

Summary:

• The car gets 30 miles to the gallon.
• After the car has traveled 80 miles, 2 and two-thirds gallons of gas have been consumed.