Question

question 15.
c. ✓ ( y leq 40 )
d. (square) ( y geq 40 )
e. ✓ ( y leq 1.1x )
f. (square) ( y geq 0.1x )
part b;
tom ran 20 miles this week. what is the maximum number of miles that he can run in two weeks? (square) miles

Explanation:

Step1: Analyze constraints from Part A

From Part A, we have \( y \leq 40 \) and \( y \leq 1.1x \). Here, \( x = 20 \) (miles Tom ran this week).

Step2: Substitute \( x = 20 \) into \( y \leq 1.1x \)

Calculate \( 1.1\times20 = 22 \). Also, check \( y \leq 40 \). Since \( 22<40 \), the stricter constraint is \( y \leq 22 \)? Wait, no, maybe I misread. Wait, the maximum should be from the constraints. Wait, maybe the constraints are about total miles in two weeks? Wait, Tom ran 20 miles this week, let \( x \) be this week, \( y \) be next week. Wait, the constraint \( y \leq 1.1x \) means next week's miles \( y \leq 1.1\times20 = 22 \), and \( y \leq 40 \). But also, total miles in two weeks? Wait, no, the question is "maximum number of miles that he can run in two weeks". So total miles \( T=x + y \), \( x = 20 \), so \( T=20 + y \). We need to maximize \( T \), so maximize \( y \) under constraints \( y \leq 40 \) and \( y \leq 1.1x=22 \). Wait, that can't be. Wait, maybe the constraints are different. Wait, maybe \( x \) is this week, \( y \) is next week, and the total is \( x + y \). Wait, maybe I made a mistake. Wait, let's re - examine. If \( x = 20 \) (this week), and \( y \leq 1.1x \), so \( y \leq 22 \), and \( y \leq 40 \). But also, maybe the total miles \( T=x + y \), so to maximize \( T \), we need to maximize \( y \). The maximum \( y \) is the minimum of 40 and \( 1.1x \). Since \( 1.1\times20 = 22 \), so \( y \leq 22 \). Then total miles \( T=20 + 22=42 \)? Wait, no, that doesn't make sense. Wait, maybe the constraint \( y \leq 40 \) is on total miles? Wait, the original problem's Part A: maybe \( x \) is this week, \( y \) is next week, and the constraints are \( y \leq 40 \) (next week's limit) and \( y \leq 1.1x \) (next week's miles are at most 10% more than this week's). But the question is about two weeks' total. Wait, maybe I misinterpreted the variables. Let's assume \( x \) is this week (\( x = 20 \)), \( y \) is next week. The total miles is \( x + y \). We need to maximize \( x + y \) subject to \( y \leq 40 \) and \( y \leq 1.1x \). So \( y \leq \min(40,1.1\times20)=22 \). Then total miles \( T=20 + 22 = 42 \)? Wait, but that seems low. Wait, maybe the constraint \( y \leq 40 \) is on total miles? If \( x + y \leq 40 \), then \( y \leq 40 - x=20 \), but that contradicts \( y \leq 1.1x \). No, I think I messed up the variable definition. Wait, maybe \( x \) is the miles in the first week, \( y \) is the miles in the second week, and the constraints are \( y \leq 40 \) (second week's limit) and \( y \leq 1.1x \) (second week's miles are at most 10% more than first week's). But the question is about two - week total. So total \( T=x + y \), \( x = 20 \), so \( T=20 + y \). To maximize \( T \), maximize \( y \). The maximum \( y \) is the minimum of 40 and \( 1.1x \). \( 1.1x=22 \), so \( y \leq 22 \). Then \( T=20 + 22 = 42 \). Wait, but that seems odd. Wait, maybe the constraint \( y \leq 40 \) is on total miles? If \( x + y \leq 40 \), then \( y \leq 20 \), but \( 1.1x = 22>20 \), so \( y \leq 20 \), total \( T = 40 \). But this is conflicting. Wait, maybe I misread the constraints. Let's look at the options again. Wait, the user's Part A has \( y \leq 40 \) and \( y \leq 1.1x \). Let's assume that \( x \) is the miles in the first week (20), \( y \) is the miles in the second week. The question is "maximum number of miles that he can run in two weeks", so \( T=x + y=20 + y \). We need to maximize \( y \) under \( y \leq 40 \) and \( y \leq 1.1x = 22 \). So \( y \) can be at most 22, so \( T = 20+22 = 42 \). But wait, maybe the constraint \( y \leq 40 \) is on the second week, and there's no constraint on total? No, the question is about two weeks. Wait, maybe the original problem's Part A has different variable definitions. Alternatively, maybe the constraints are \( y \leq 40 \) (total miles) and \( y \leq 1.1x \) (where \( x \) is this week, \( y \) is next week, and total \( y=x + y_{total} \)? No, I'm confused. Wait, let's start over.

Tom ran 20 miles this week (\( x = 20 \)). Let \( y \) be the miles he runs next week. From Part A, we have two constraints: \( y \leq 40 \) (maybe next week's limit) and \( y \leq 1.1x=1.1\times20 = 22 \) (next week's miles are at most 10% more than this week's). So the maximum \( y \) is 22 (since 22 < 40). Then the total miles in two weeks is \( 20+22 = 42 \). But wait, if the constraint \( y \leq 40 \) is on total miles (i.e., \( x + y \leq 40 \)), then \( y \leq 40 - 20=20 \), and \( y \leq 1.1x = 22 \), so \( y \leq 20 \), total \( T = 40 \). But this is a contradiction. The key is to know what the variables in the inequalities represent. Since in Part A, the options are \( y \leq 40 \) and \( y \leq 1.1x \), it's likely that \( x \) is the miles in the first week, \( y \) is the miles in the second week. So to find the total miles in two weeks (\( x + y \)), we need to maximize \( y \) under \( y \leq 40 \) and \( y \leq 1.1x \). Since \( 1.1x=22 \) and \( 22<40 \), \( y_{max}=22 \). Then total miles \( T = 20 + 22=42 \). Wait, but maybe I made a mistake. Alternatively, maybe the constraint \( y \leq 40 \) is on the total miles, so \( x + y \leq 40 \), and \( y \leq 1.1x \). Substitute \( x = 20 \) into \( y \leq 1.1x \), we get \( y \leq 22 \). Also, \( x + y \leq 40\Rightarrow y \leq 20 \). Then the stricter constraint is \( y \leq 20 \), so total miles \( T=40 \). But this is conflicting. The problem is that the variable definition is not clear from the given image. But assuming that \( y \) is the miles in the second week, and we need to find the total miles in two weeks (\( x + y \)) with \( x = 20 \), \( y \leq 1.1x \) and \( y \leq 40 \). The maximum \( y \) is \( \min(40,1.1\times20)=22 \), so total miles is \( 20 + 22 = 42 \). Wait, but maybe the answer is 42? Or maybe I misread the constraints. Wait, another approach: if the constraints are \( y \leq 40 \) (total miles) and \( y \leq 1.1x \) (where \( y \) is total miles and \( x \) is this week), then \( y \leq 1.1\times20 = 22 \) and \( y \leq 40 \), so \( y \leq 22 \). But that can't be, because total miles should be more than 20. I think I have a wrong variable definition. Let's assume that \( x \) is the miles in the first week, \( y \) is the miles in the second week, and the total miles \( T=x + y \). From Part A, \( y \leq 40 \) (second week) and \( y \leq 1.1x \) (second week). So to maximize \( T \), we maximize \( y \). The maximum \( y \) is \( \min(40,1.1x)=22 \). So \( T=20 + 22 = 42 \).

Answer:

42