Question

tom is training to run a race. he wants to run no more than 40 miles per week and increase the weekly distance that he runs by no more than 10% per week.
part a:
if x represents the number of miles that tom ran during the previous week, which of the following equations represent the number of miles (y) that tom can run week? select all that apply.
a. $square$ $y \leq 40 + 1.1x$
b. $square$ $y \leq 40 + 0.1x$
c. $square$ $y \leq 40$
d. $square$ $y \geq 40$
e. $square$ $y \leq 1.1x$

Explanation:

Step1: Analyze the 10% increase

A 10% increase on \( x \) means the new distance from the increase is \( x + 0.1x=1.1x \). But Tom also can't exceed 40 miles per week. Wait, no—wait, the problem says "run no more than 40 miles per week and increase the weekly distance that he runs by no more than 10% per week". Wait, maybe I misread. Wait, actually, the two constraints: total miles \( y \leq 40 \), and also \( y \leq 1.1x \) (since increasing by no more than 10% means \( y \leq x + 0.1x = 1.1x \)). Wait, but let's check the options. Wait, maybe the problem is that the 40 is a cap, and the increase is on the previous week. Wait, maybe the options have a typo? Wait, no, let's re-express:

Tom's weekly distance \( y \) must satisfy two things:

1. \( y \leq 40 \) (no more than 40 miles per week)
2. \( y \leq 1.1x \) (increase by no more than 10% from previous week \( x \))

Wait, but the options given:

A. \( y \leq 40 + 1.1x \) – that doesn't make sense, because 40 + 1.1x would be way more than 40.

B. \( y \leq 40 + 0.1x \) – also doesn't make sense.

C. \( y \leq 40 \) – that's one constraint.

E. \( y \leq 1.1x \) – that's the other constraint (since 10% increase is \( 1.1x \)).

Wait, maybe the problem was misphrased, and the "no more than 40" and "increase by no more than 10%", so the correct inequalities are \( y \leq 40 \) and \( y \leq 1.1x \). So looking at the options, C and E (if E is \( y \leq 1.1x \)) would apply. But let's check the original problem again. Wait, the user's image shows options:

A. \( y \leq 40 + 1.1x \)

B. \( y \leq 40 + 0.1x \)

C. \( y \leq 40 \)

D. \( y \geq 40 \)

E. \( y \leq 1.1x \) (assuming E is \( y \leq 1.1x \))

So the correct ones are C (since he runs no more than 40 miles) and E (since he increases by no more than 10%, so \( y \leq x + 0.1x = 1.1x \)). Wait, but maybe the problem's options have a mistake, but based on the constraints, \( y \leq 40 \) (option C) and \( y \leq 1.1x \) (option E, if that's the case). But let's check again.

Wait, maybe I misinterpreted the problem. Maybe the "no more than 40 miles per week" and "increase the weekly distance by no more than 10%", so the maximum \( y \) is the minimum of 40 and 1.1x. So the inequalities are \( y \leq 40 \) and \( y \leq 1.1x \). So among the options, C (\( y \leq 40 \)) and E (\( y \leq 1.1x \)) are correct. But let's check the options given. If E is \( y \leq 1.1x \), then C and E. But maybe the original problem had a different setup. Wait, maybe the 40 is not a hard cap, but the increase is on top of 40? No, that doesn't make sense. Alternatively, maybe the problem is that the two constraints are: \( y \leq 40 \) and \( y \leq x + 0.1x \), so \( y \leq 1.1x \). So the correct options are C and E. But let's confirm:

• Option C: \( y \leq 40 \) – correct, since he wants to run no more than 40 miles per week.

• Option E: \( y \leq 1.1x \) – correct, since increasing by no more than 10% means \( y \leq x + 0.1x = 1.1x \).

Options A and B add 40 to the increased distance, which is illogical. Option D is the opposite of C. So the correct options are C and E (assuming E is \( y \leq 1.1x \)). But since the user's image shows E as \( y \leq 1.1x \) (partially visible), so we'll go with C and E. But wait, the problem says "Select ALL that apply". So:

• C: \( y \leq 40 \) – correct.

• E: \( y \leq 1.1x \) – correct.

But let's check again. Maybe the problem was intended to have \( y \leq \min(40, 1.1x) \), so both inequalities must hold. So the correct options are C and E.

Answer:

C. \( y \leq 40 \), E. \( y \leq 1.1x \) (assuming E is \( y \leq 1.1x \) as per the image)

But since the options are labeled A to E, and E is \( y \leq 1.1x \), the correct options are C and E. However, if we have to choose from the given options (as per the image), the correct ones are C (\( y \leq 40 \)) and E (\( y \leq 1.1x \)).