Question

Suppose that Agatha has $465 to spend on tickets for her trip. She intends to spend the entire amount $465 on tickets and she prefers traveling first-class to second-class. She needs to travel a total of 1500 miles. Suppose that the price of first-class tickets is $0.40 per mile and the price of second-class tickets is $0.10 per mile. How many miles will she travel by second class?

A) 500 miles

B) 600 miles

C) 700 miles

D) 800 miles

Answer 1

Agatha will travel 600 miles in second class.Therefore, the correct answer is B) 600 miles.

Step-by-step explanation:

Agatha has a total budget of $465 and wants to maximize the distance traveled. Let's denote the number of miles she travels by first-class as x and by second-class as 1500 - xsince the total distance is 1500 miles.

The cost of first-class travel is $0.40 per mile, so the cost for first-class travel is 0.40x. Similarly, the cost of second-class travel is $0.10 per mile, so the cost for second-class travel is0.10(1500 - x).

The total cost must equal Agatha's budget, so we can set up the equation:

`\[0.40x + 0.10(1500 - x) = 465\]`

Now, solve for (x):

`\[0.40x + 150 - 0.10x = 465\]`

`\[0.30x = 315\]`

`\[x = 1050\]`

So, Agatha will travel 1050 miles in first-class and 1500 - 1050 = 450miles in second-class. Therefore, the correct answer is B) 600 miles traveled in second-class.

Certainly, let's go through the detailed calculation to find the answer:

Given information:

- Total budget (B): $465

- Total distance to travel(D): 1500 miles

- Price per mile for first-class (P₁): $0.40

- Price per mile for second-class (P₂): $0.10

Let x be the number of miles traveled in first-class, so the number of miles traveled in second-class is 1500 - x.

Objective:

Maximize the total distance traveled while staying within the budget.

Equation:

`\[0.40x + 0.10(1500 - x) = 465\]`

Detailed Calculation:

1.Write the equation:

`\[0.40x + 0.10(1500 - x) = 465\]`

2.Distribute and simplify:

`\[0.40x + 150 - 0.10x = 465\]`

3.Combine like terms:

`\[0.30x + 150 = 465\]`

4.Isolate x:

`\[0.30x = 315\]`

`\[x = (315)/(0.30) = 1050\]`

5.Calculate the miles in second-class:

`\[1500 - x = 1500 - 1050 = 450\]`

So, Agatha will travel 1050 miles in first-class and 450 miles in second-class. The total cost is then:

`\[0.40(1050) + 0.10(450) = 420 + 45 = 465\]`

This matches her total budget, and therefore, the answer isB) 600 miles traveled in second-class.