Question

#12 modeling real life a one-year-old boa constrictor is 26 inches long and grows 8 inches per year. write an equation in slope-intercept form that represents the length y (in feet) of a boa constrictor that is x years old. y = \square

Explanation:

Step1: Identify slope and intercept

The slope-intercept form is \( y = mx + b \), where \( m \) is the slope (rate of growth) and \( b \) is the y-intercept (initial length). The boa grows 8 inches per year, so \( m = 8 \). The initial length (when \( x = 0 \), but here \( x = 1 \) year old is 26 inches? Wait, no—wait, the problem says a one-year-old is 26 inches. Wait, maybe \( x = 0 \) is birth? Wait, no, the problem says "a one-year-old boa constrictor is 26 inches long and grows 8 inches per year". So when \( x = 1 \), \( y = 26 \)? Wait, no, maybe the initial length (at \( x = 0 \), which would be 0 years old, but the boa is one-year-old at 26 inches. Wait, maybe the problem is that at \( x = 0 \) (birth), the length is \( 26 - 8 = 18 \) inches? No, that doesn't make sense. Wait, no—wait, the problem says "a one-year-old boa constrictor is 26 inches long and grows 8 inches per year". So the growth rate is 8 inches per year, so the slope \( m = 8 \). The y-intercept \( b \) is the length when \( x = 0 \) (0 years old). But if at \( x = 1 \) (1 year old), it's 26 inches, then \( 26 = 8(1) + b \), so \( b = 26 - 8 = 18 \). Wait, but maybe the problem is considering \( x = 0 \) as 1 year old? No, that's confusing. Wait, no—maybe the problem has a typo, but actually, the standard slope-intercept: the initial length (when \( x = 0 \)) is 26 inches? Wait, no, the boa is one-year-old at 26 inches. Wait, maybe the problem means that at \( x = 0 \) (0 years old), the length is 26 inches, and it grows 8 inches per year. But the problem says "a one-year-old" is 26 inches. Hmm. Wait, maybe the problem is that the initial length (b) is 26 inches, and the slope (m) is 8 inches per year, with \( x \) being the number of years since it was one-year-old? No, that's overcomplicating. Wait, the problem says "Write an equation in slope-intercept form that represents the length \( y \) (in inches) of a boa constrictor that is \( x \) years old." Wait, the original problem's y is in feet? Wait, no, the image says "y (in feet)"? Wait, no, the text says "y (in feet)"? Wait, the image: "y (in feet) of a boa constrictor that is x years old. A one-year-old boa constrictor is 26 inches long and grows 8 inches per year. Write an equation in slope-intercept form..." Wait, but y is in feet, so we need to convert inches to feet. Oh! That's the key. 26 inches is \( \frac{26}{12} \) feet, and 8 inches per year is \( \frac{8}{12} = \frac{2}{3} \) feet per year. Wait, no—wait, the problem says "y (in feet)"? Wait, the image: "y (in feet)"? Wait, the user's image: "y (in feet) of a boa constrictor that is x years old. A one-year-old boa constrictor is 26 inches long and grows 8 inches per year. Write an equation in slope-intercept form..." So we need to convert inches to feet.

First, convert initial length: 26 inches to feet: \( 26 \div 12 = \frac{13}{6} \) feet? Wait, no, wait—the boa is one-year-old (x=1) with length 26 inches. Wait, no, maybe the initial length (at x=0, which is 0 years old) is 26 inches? No, the problem says "a one-year-old" is 26 inches. So when x=1, y (in inches) is 26, and the growth rate is 8 inches per year. But the problem says y is in feet. So we need to convert the units.

Wait, let's re-express:

- Growth rate: 8 inches per year = \( \frac{8}{12} = \frac{2}{3} \) feet per year.

- Initial length (when x=0, 0 years old): but the boa is one-year-old at 26 inches. Wait, maybe the problem has x=0 as 1 year old? No, that's not standard. Wait, maybe the problem is that the initial length (b) is 26 inches (in feet, \( \frac{26}{12} = \frac{13}{6} \)) and the slope (m) is 8 inches per year (in feet, \( \frac{8}{12} = \frac{2}{3} \)). But then the equation would be \( y = \frac{2}{3}x + \frac{13}{6} \)? No, that seems complicated. Wait, maybe the problem has a mistake, and y is in inches, not feet. Let's check the image again. The image says "y (in feet)"? Wait, the user's image: "y (in feet) of a boa constrictor that is x years old. A one-year-old boa constrictor is 26 inches long and grows 8 inches per year. Write an equation in slope-intercept form..." So we need to convert inches to feet.

26 inches = \( \frac{26}{12} = \frac{13}{6} \) feet ≈ 2.1667 feet.

8 inches per year = \( \frac{8}{12} = \frac{2}{3} \) feet per year ≈ 0.6667 feet per year.

But wait, if the boa is one-year-old (x=1) with length 26 inches ( \( \frac{13}{6} \) feet), then the equation would be \( y = \frac{2}{3}x + b \). Plugging x=1, y= \( \frac{13}{6} \):

\( \frac{13}{6} = \frac{2}{3}(1) + b \)

\( b = \frac{13}{6} - \frac{4}{6} = \frac{9}{6} = \frac{3}{2} \) feet (which is 18 inches). So the equation is \( y = \frac{2}{3}x + \frac{3}{2} \). But that seems odd. Wait, maybe the problem meant y is in inches, not feet. Let's assume that (maybe a typo in the image). Then:

Initial length (b) = 26 inches (when x=1? No, that can't be. Wait, no—if the boa is one-year-old (x=1) and is 26 inches, and grows 8 inches per year, then the equation in inches would be \( y = 8x + b \). When x=1, y=26: 26 = 8(1) + b → b=18. So the equation in inches is \( y = 8x + 18 \). But the problem says y is in feet. So convert 8 inches to feet: \( \frac{8}{12} = \frac{2}{3} \) feet per year, and 18 inches to feet: \( \frac{18}{12} = \frac{3}{2} \) feet. So the equation in feet is \( y = \frac{2}{3}x + \frac{3}{2} \). But maybe the problem has x as the number of years since it was one-year-old? No, that's not likely. Wait, maybe the initial length is 26 inches (at x=0, which is 0 years old), and it grows 8 inches per year. Then in inches, \( y = 8x + 26 \), and in feet, \( y = \frac{8}{12}x + \frac{26}{12} = \frac{2}{3}x + \frac{13}{6} \). But the problem says "a one-year-old" is 26 inches, so when x=1, y=26. So with \( y = 8x + b \), 26 = 8(1) + b → b=18. So in inches, \( y = 8x + 18 \). Converting to feet: \( y = \frac{8}{12}x + \frac{18}{12} = \frac{2}{3}x + \frac{3}{2} \). But maybe the problem intended y to be in inches, so the equation is \( y = 8x + 18 \). Wait, the image's y is in feet, but maybe that's a mistake. Alternatively, maybe the initial length is 26 inches (at x=0, 0 years old), and it grows 8 inches per year, so at x=1 (1 year old), it's 34 inches. But the problem says 26 inches at one year old. So there's a contradiction. Wait, maybe the problem is that the boa is one-year-old (x=1) with length 26 inches, so the growth rate is 8 inches per year, so the equation is \( y = 8(x - 1) + 26 \), which simplifies to \( y = 8x - 8 + 26 = 8x + 18 \). Ah, that makes sense! So when x=1, y=8(1)+18=26, which matches. So the slope is 8 (inches per year), and the y-intercept is 18 (inches, the length at x=0, which is 0 years old, i.e., birth). Then, since y is in feet, we convert 8 inches to feet: \( \frac{8}{12} = \frac{2}{3} \) feet per year, and 18 inches to feet: \( \frac{18}{12} = \frac{3}{2} \) feet. So the equation in feet is \( y = \frac{2}{3}x + \frac{3}{2} \). But maybe the problem wants the answer in inches, ignoring the "in feet" (maybe a typo). Let's proceed with inches first.

Step2: Write the equation

Using slope-intercept form \( y = mx + b \), where \( m = 8 \) (inches per year) and \( b = 18 \) (inches at x=0, since at x=1, 8(1)+18=26). So the equation is \( y = 8x + 18 \). If we need to convert to feet, \( y = \frac{8}{12}x + \frac{18}{12} = \frac{2}{3}x + \frac{3}{2} \). But the problem's y is in feet, so we'll use that. Wait, but 8 inches per year is \( \frac{2}{3} \) feet per year, and initial length (at x=0) is 18 inches = \( \frac{3}{2} \) feet. So the equation is \( y = \frac{2}{3}x + \frac{3}{2} \). But maybe the problem intended y to be in inches, so \( y = 8x + 18 \). Given the confusion, but the key is slope-intercept: rate of growth is 8 inches per year (slope), initial length (at x=0) is 18 inches (since at x=1, 26=8+18), so equation is \( y = 8x + 18 \) (in inches) or \( y = \frac{2}{3}x + \frac{3}{2} \) (in feet). But the problem says "y (in feet)", so we'll use the feet version. Wait, but 8 inches is \( \frac{2}{3} \) feet, and 18 inches is \( \frac{3}{2} \) feet. So \( y = \frac{2}{3}x + \frac{3}{2} \). Alternatively, maybe the problem has a mistake and y is in inches, so \( y = 8x + 18 \). Let's check the problem again: "a one-year-old boa constrictor is 26 inches long and grows 8 inches per year". So when x=1, y=26 (inches). So the equation is \( y = 8x + 18 \) (since 8(1)+18=26). So in inches, that's correct. If we convert to feet, \( y = \frac{8}{12}x + \frac{18}{12} = \frac{2}{3}x + \frac{3}{2} \). But the problem says y is in feet, so we'll present that.

Answer:

\( y = \frac{2}{3}x + \frac{3}{2} \) (or if in inches, \( y = 8x + 18 \); but based on y in feet, \( \frac{2}{3}x + \frac{3}{2} \))