College Algebra Word Problem Guidance: Distance, Rate, and Geometry

Q1. Two cars 120 miles apart begin driving towards each other. One car travels at 20 mph, the other at 40 mph. At the same time, a canary starts on one car and flies back and forth between the two cars at 150 mph until the cars meet. How far has the canary flown when the cars meet?

Background

Topic: Distance, Rate, and Time Problems

This question tests your ability to use the relationship between distance, rate, and time to solve a classic "canary and cars" problem, often called the "fly problem." It requires setting up equations for the time until the cars meet and then using that time to determine the total distance the canary travels.

Key Terms and Formulas

• Distance formula: $\text{Distance} = \text{Rate} \times \text{Time}$

• Relative speed: When two objects move towards each other, their speeds add.

Step-by-Step Guidance

1. First, determine how long it takes for the two cars to meet. Since they are moving towards each other, add their speeds: $20 \text{ mph} + 40 \text{ mph} = 60 \text{ mph}$.

2. Set up the equation for the time until they meet: $\text{Time} = \frac{\text{Distance}}{\text{Combined Rate}} = \frac{120}{60}$.

3. Once you have the time, use the canary's speed to find the total distance it flies: $\text{Distance flown} = 150 \text{ mph} \times \text{Time}$.

4. Plug in the value for time from step 2 into the equation in step 3, but do not calculate the final value yet.

Try solving on your own before revealing the answer!

Q2. Juan is decorating for a party in a room with ten large cylindrical posts, each 8 feet high and with a circumference of 6 ft. He plans to wrap eight turns of ribbon around each post. How much ribbon does Juan need?

Background

Topic: Geometry and Measurement

This question tests your ability to calculate the total length of ribbon needed by considering the number of posts, the number of turns, and the circumference of each post.

Key Terms and Formulas

• Circumference of a cylinder: $\text{Circumference} = 2\pi r$ (but here, the circumference is given as 6 ft)

• Total ribbon per post: $\text{Ribbon per post} = \text{Number of turns} \times \text{Circumference}$

• Total ribbon needed: $\text{Total ribbon} = \text{Ribbon per post} \times \text{Number of posts}$

Step-by-Step Guidance

1. Calculate the ribbon needed for one post: $8 \text{ turns} \times 6 \text{ ft} =$ (do not compute yet).

2. Multiply the result from step 1 by the number of posts: $\text{Result from step 1} \times 10$.

3. Set up the final multiplication, but do not compute the total length yet.

Try solving on your own before revealing the answer!

Q3. Jordan and Amari run a 200-meter race and Jordan wins by 10 meters. They decide to run the 200 meter race again with Jordan starting 10 meters behind the starting line.

Part a) Assuming both runners run at the same pace as they did in the first race, who wins the second race?

Background

Topic: Proportional Reasoning and Rates

This question tests your understanding of relative speed and how a head start or handicap affects the outcome of a race.

Key Terms and Formulas

• Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

• Relative performance: Compare the time each runner takes to complete their respective distances.

Step-by-Step Guidance

1. From the first race, Jordan covers 200 meters while Amari covers 190 meters in the same time. This means Jordan's speed is $\frac{200}{t}$ and Amari's is $\frac{190}{t}$, where $t$ is the time it takes Jordan to finish.

2. In the second race, Jordan starts 10 meters behind, so he must run 210 meters, while Amari runs 200 meters.

3. Set up equations for the time each runner takes: $\text{Time for Jordan} = \frac{210}{\text{Jordan's speed}}$, $\text{Time for Amari} = \frac{200}{\text{Amari's speed}}$.

4. Compare the two times to determine who finishes first, but do not compute the final comparison yet.

Try solving on your own before revealing the answer!

Part b) How far behind the starting line must Jordan start so the second race is a tie?

Background

Topic: Setting up Equations for Equal Outcomes

This question tests your ability to set up and solve an equation where two runners finish at the same time, given their speeds and distances.

Key Terms and Formulas

• Let $x$ be the distance Jordan starts behind the line.

• Set up the equation: $\frac{200 + x}{\text{Jordan's speed}} = \frac{200}{\text{Amari's speed}}$

Step-by-Step Guidance

1. Recall from the first race: Jordan's speed is $\frac{200}{t}$, Amari's is $\frac{190}{t}$.

2. Set up the equation for a tie: $\frac{200 + x}{\text{Jordan's speed}} = \frac{200}{\text{Amari's speed}}$.

3. Substitute the speeds from the first race into the equation.

4. Solve for $x$, but stop before the final calculation.

Try solving on your own before revealing the answer!