Question

when carrie gets home from school, she turns the heat on to warm the house. over time, the temperature in the house rises. this situation can be modeled as a linear relationship. complete the statement that describes the situation. carries house is dropdown when she gets home. after she turns the heat on, the temperature in the house increases by dropdown.

Explanation:

Step1: Find initial temperature

When Carrie gets home, heating time \( x = 0 \). From the graph, at \( x = 0 \), \( y = 64^\circ\text{F} \). So initial temperature is \( 64^\circ\text{F} \).

Step2: Calculate temperature increase rate

Pick two points, e.g., \( (0, 64) \) and \( (15, 65) \). Slope \( m=\frac{65 - 64}{15 - 0}=\frac{1}{15}\approx0.0667^\circ\text{F per minute} \)? Wait, no, wait. Wait, let's check another point. Wait, at \( x = 15 \), \( y = 65 \); \( x = 30 \), \( y = 66 \). So from \( x = 0 \) to \( x = 15 \), change in \( x = 15 \), change in \( y = 1 \). So rate is \( \frac{1}{15} \)? Wait, no, wait the grid: each x - axis grid is 5 minutes? Wait, x - axis: 0, 5, 10, 15, 20,... So from \( x = 0 \) (y = 64) to \( x = 15 \) (y = 65): 15 minutes, temperature increases by 1°F. So per 15 minutes, 1°F, or per minute \( \frac{1}{15} \)? Wait, no, maybe I misread. Wait, the first blank is initial temperature (when x = 0, y = 64), so first blank: \( 64^\circ\text{F} \). Then the rate: let's take two points: (0, 64) and (15, 65). The change in temperature is \( 65 - 64 = 1^\circ\text{F} \), change in time is \( 15 - 0 = 15 \) minutes. Wait, but maybe another way: from x = 0 to x = 30, y goes from 64 to 66: 30 minutes, 2°F, so rate \( \frac{2}{30}=\frac{1}{15}\approx0.0667 \)? No, wait the problem says "increases by" – maybe per 15 minutes 1°F, or per minute? Wait, no, let's check the graph again. At x = 0, y = 64; x = 15, y = 65; x = 30, y = 66; x = 45, y = 67. So every 15 minutes, temperature increases by 1°F. So the rate is \( 1^\circ\text{F} \) every 15 minutes, or \( \frac{1}{15}^\circ\text{F per minute} \). But maybe the problem expects the initial temperature as 64°F, and the increase rate: let's see, the first blank is the initial temperature (when she gets home, x = 0), so 64°F. Then, after turning heat on, temperature increases by, say, 1°F every 15 minutes? Wait, no, maybe the problem is simpler: when x = 0, y = 64, so first blank: 64 degrees Fahrenheit. Then, the slope: from (0,64) to (15,65), the change in y is 1, change in x is 15, so the rate is \( \frac{1}{15}^\circ\text{F per minute} \)? Wait, no, maybe the problem has the first blank as 64°F, and the second blank: let's see, the options? Wait, the problem has dropdowns. But from the graph, initial temperature (x = 0) is 64°F. Then, the temperature increase: let's take two points, (0,64) and (15,65). So over 15 minutes, temperature increases by 1°F. So the rate is 1°F every 15 minutes, or \( \frac{1}{15}^\circ\text{F per minute} \). But maybe the problem expects the initial temperature as 64°F, and the increase rate as \( \frac{1}{15}^\circ\text{F} \) per minute? Wait, no, maybe I made a mistake. Wait, the first part: "Carrie's house is [blank] when she gets home" – that's the y - intercept, which is 64°F. Then, "after she turns the heat on, the temperature in the house increases by [blank]". Let's calculate the slope. The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Take (0,64) and (15,65): \( m=\frac{65 - 64}{15 - 0}=\frac{1}{15}\approx0.0667^\circ\text{F per minute} \). But maybe the problem has the first blank as 64°F, and the second blank as \( \frac{1}{15}^\circ\text{F} \) per minute, or 1°F every 15 minutes. But from the graph, at x = 0, y = 64, so first blank is 64°F. Then, the increase rate: let's check the difference between x = 0 and x = 15: 15 minutes, temperature goes from 64 to 65, so increase by 1°F over 15 minutes, so rate is \( \frac{1}{15}^\circ\text{F} \) per minute, or 1°F every 15 minutes. But the problem says "increases by" – maybe the unit is per minute or per 15 minutes. But from the graph, the first value is 64°F, and the rate is \( \frac{1}{15}^\circ\text{F} \) per minute (or 1°F every 15 minutes). But let's confirm: at x = 0, y = 64; x = 15, y = 65; x = 30, y = 66; x = 45, y = 67. So every 15 minutes, temperature increases by 1°F. So the rate is 1°F per 15 minutes, or \( \frac{1}{15}^\circ\text{F} \) per minute.

Answer:

First blank: \( 64^\circ\text{F} \), second blank: \( \frac{1}{15}^\circ\text{F} \) per minute (or 1°F every 15 minutes). But from the graph, initial temperature is \( 64^\circ\text{F} \), and the temperature increases by \( \frac{1}{15}^\circ\text{F} \) per minute (or 1°F every 15 minutes). So the first blank is \( 64^\circ\text{F} \), the second blank: let's see, the slope is \( \frac{1}{15} \), so the temperature increases by \( \frac{1}{15}^\circ\text{F} \) each minute, or 1°F every 15 minutes.