Calculating the Distance Traveled by Summer

Oct 13, 2025 by recoveryshake.com 145 views

In mathematics, speed and time are two fundamental concepts that are often used to solve problems involving motion. In this article, we will explore how to use these concepts to solve a problem involving two individuals, Summer and Trinity, who are walking at different speeds.

The Problem

Summer started walking 30 seconds before Trinity. Summer walked at a speed of 3 inches per second, while Trinity walked at a speed of 8 inches per second. After a total of x seconds, how far apart will Summer and Trinity be?

Breaking Down the Problem

To solve this problem, we need to break it down into smaller, more manageable parts. Let's start by analyzing the situation:

Summer starts walking 30 seconds before Trinity.
Summer walks at a speed of 3 inches per second.
Trinity walks at a speed of 8 inches per second.

Calculating the Distance Traveled by Summer

Since Summer starts walking 30 seconds before Trinity, we need to calculate the distance traveled by Summer in those 30 seconds. We can do this by multiplying the speed of Summer (3 inches per second) by the time (30 seconds):

Distance traveled by Summer = Speed x Time = 3 inches/second x 30 seconds = 90 inches

Calculating the Distance Traveled by Trinity

Now that we know the distance traveled by Summer, we need to calculate the distance traveled by Trinity. We can do this by multiplying the speed of Trinity (8 inches per second) by the time (x seconds):

Distance traveled by Trinity = Speed x Time = 8 inches/second x x seconds = 8x inches

Calculating the Total Distance Between Summer and Trinity

Since Summer and Trinity are walking in the same direction, the total distance between them will be the sum of the distances traveled by each of them. We can calculate this by adding the distance traveled by Summer (90 inches) to the distance traveled by Trinity (8x inches):

Total distance = Distance traveled by Summer + Distance traveled by Trinity = 90 inches + 8x inches = 90 + 8x inches

Simplifying the Equation

Now that we have the total distance between Summer and Trinity, we can simplify the equation by combining like terms:

Total distance = 90 + 8x inches

Finding the Value of x

Since we are asked to find the value of x, we need to isolate x on one side of the equation. We can do this by subtracting 90 from both sides of the equation:

8x = Total distance - 90 8x = 8x

Dividing Both Sides by 8

Now that we have isolated x, we can divide both sides of the equation by 8 to find the value of x:

x = (Total distance - 90) / 8

Substituting the Value of Total Distance

Since we are asked to find the value of x after a total of x seconds, we can substitute the value of total distance (90 + 8x inches) into the equation:

x = (90 + 8x - 90) / 8 x = 8x / 8 x = x

Simplifying the Equation

Now that we have substituted the value of total distance, we can simplify the equation by combining like terms:

x = x

Finding the Value of x

Since we are asked to find the value of x, we need to isolate x on one side of the equation. We can do this by subtracting x from both sides of the equation:

0 = x - x 0 = 0

Dividing Both Sides by 0

Now that we have isolated x, we can divide both sides of the equation by 0 to find the value of x:

x = 0 / 0

Undefined Value

Since we are dividing by 0, the value of x is undefined.

Conclusion

In this article, we will answer some of the most frequently asked questions related to the problem of Summer and Trinity walking at different speeds.

Q: What is the problem about?

A: The problem is about two individuals, Summer and Trinity, who are walking at different speeds. Summer starts walking 30 seconds before Trinity and walks at a speed of 3 inches per second, while Trinity walks at a speed of 8 inches per second. We need to find the distance between them after a total of x seconds.

Q: Why is the value of x undefined?

A: The value of x is undefined because we are dividing by 0. When we subtract x from both sides of the equation, we get 0 = x - x, which simplifies to 0 = 0. Then, when we divide both sides by 0, we get x = 0 / 0, which is undefined.

Q: What does it mean to divide by 0?

A: Dividing by 0 is undefined in mathematics because it does not make sense to divide a number by zero. Think of it like this: if you have a pizza that is divided into 4 slices, and you want to divide it into 0 slices, how many slices would you have? The answer is not a number, it's undefined.

Q: Can we solve the problem in a different way?

A: Yes, we can solve the problem in a different way. Instead of using algebraic equations, we can use a different approach. We can calculate the distance traveled by Summer and Trinity separately and then find the difference between their distances.

Q: How can we calculate the distance traveled by Summer and Trinity?

A: We can calculate the distance traveled by Summer and Trinity by multiplying their speeds by the time they walked. For Summer, the distance traveled is 3 inches/second x x seconds = 3x inches. For Trinity, the distance traveled is 8 inches/second x x seconds = 8x inches.

Q: How can we find the distance between Summer and Trinity?

A: We can find the distance between Summer and Trinity by subtracting the distance traveled by Summer from the distance traveled by Trinity. This gives us:

Distance between Summer and Trinity = Distance traveled by Trinity - Distance traveled by Summer = 8x inches - 3x inches = 5x inches

Q: What is the value of x?

A: Unfortunately, we still can't find the value of x because we don't have enough information. We need to know the time that Summer and Trinity walked for, which is represented by x.

Q: Can we use a different approach to find the value of x?

A: Yes, we can use a different approach to find the value of x. We can use the fact that Summer started walking 30 seconds before Trinity. This means that Summer walked for x + 30 seconds, while Trinity walked for x seconds.

Q: How can we use this information to find the value of x?

A: We can use this information to find the value of x by setting up an equation. We know that the distance traveled by Summer is 3 inches/second x (x + 30) seconds, and the distance traveled by Trinity is 8 inches/second x seconds. We can set up an equation using these two expressions and solve for x.

Q: What is the equation?

A: The equation is:

3(x + 30) = 8x

Q: How can we solve this equation?

A: We can solve this equation by expanding the left-hand side and then simplifying the equation. This gives us:

3x + 90 = 8x

Q: How can we isolate x?

A: We can isolate x by subtracting 3x from both sides of the equation. This gives us:

90 = 5x

Q: How can we find the value of x?

A: We can find the value of x by dividing both sides of the equation by 5. This gives us:

x = 18

Q: What does this mean?

A: This means that Summer and Trinity will be 5x inches apart after 18 seconds.

Conclusion

In this article, we answered some of the most frequently asked questions related to the problem of Summer and Trinity walking at different speeds. We used different approaches to solve the problem and found that the value of x is 18.