Question

suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. after 8 hours of burning, a candle has a height of 28.2 centimeters. after 26 hours of burning, its height is 26.4 centimeters. what is the height of the candle after 10 hours?

Explanation:

Step1: Find the rate of change (slope)

The slope \( m \) of a linear function is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1,y_1)=(8,28.2) \) and \( (x_2,y_2)=(26,26.4) \). So, \( m=\frac{26.4 - 28.2}{26 - 8}=\frac{- 1.8}{18}=- 0.1 \).

Step2: Find the equation of the line

Using the point - slope form \( y - y_1=m(x - x_1) \), with \( (x_1,y_1)=(8,28.2) \) and \( m=-0.1 \). We get \( y-28.2=-0.1(x - 8) \). Simplify it: \( y-28.2=-0.1x + 0.8 \), so \( y=-0.1x + 29 \).

Step3: Find the height at \( x = 10 \)

Substitute \( x = 10 \) into the equation \( y=-0.1x + 29 \). Then \( y=-0.1\times10 + 29=-1 + 29 = 28 \)? Wait, no, wait. Wait, the problem says "After 10 hours"? Wait, let's re - check the problem. Wait, the original problem: "After 8 hours of burning, a candle has a height of 28.2 centimeters. After 26 hours of burning, its height is 26.4 centimeters. What is the height of the candle after 10 hours?"

Wait, my mistake in step 2. Let's recalculate the slope. \( m=\frac{26.4 - 28.2}{26 - 8}=\frac{-1.8}{18}=-0.1 \). The equation is \( y=mx + b \). Using \( (8,28.2) \): \( 28.2=-0.1\times8 + b \), \( 28.2=-0.8 + b \), so \( b = 29 \). So the equation is \( y=-0.1x + 29 \). Now, when \( x = 10 \), \( y=-0.1\times10+29=-1 + 29 = 28 \)? Wait, but that seems odd. Wait, maybe I misread the points. Wait, the first point: after 8 hours, height 28.2. After 26 hours, height 26.4. The time difference is \( 26 - 8 = 18 \) hours, height difference is \( 26.4 - 28.2=-1.8 \) cm. So the rate of decrease is \( 0.1 \) cm per hour. Now, from 8 hours to 10 hours, that's 2 hours later. So the height at 10 hours is \( 28.2-0.1\times(10 - 8)=28.2-0.2 = 28 \) cm? Wait, but let's check with the equation. When \( x = 8 \), \( y=-0.1\times8 + 29=-0.8 + 29 = 28.2 \), correct. When \( x = 26 \), \( y=-0.1\times26+29=-2.6 + 29 = 26.4 \), correct. So when \( x = 10 \), \( y=-0.1\times10 + 29 = 28 \).

Wait, but the button shows 5 and x. Wait, maybe the problem was mis - transcribed. Wait, maybe the time is 5? Wait, no, the user's image: the button is 5 and x, and the text box is for centimeters. Wait, maybe I misread the problem. Wait, let's re - examine the problem statement.

Wait, the original problem: "Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After 8 hours of burning, a candle has a height of 28.2 centimeters. After 26 hours of burning, its height is 26.4 centimeters. What is the height of the candle after 10 hours?" Wait, but the button in the image is 5 and x, maybe the problem is after 5 hours? Wait, maybe a mis - print. Wait, let's assume that maybe the time is 5 hours. Let's recalculate.

If we want to find the height after 5 hours. Using the equation \( y=-0.1x + 29 \). When \( x = 5 \), \( y=-0.1\times5+29=-0.5 + 29 = 28.5 \). But the button is 5 and x. Wait, maybe the problem is: after 8 hours, 28.2; after 26 hours, 26.4. Find the height after 5 hours?

Wait, let's recast. The slope \( m=\frac{26.4 - 28.2}{26 - 8}=\frac{-1.8}{18}=-0.1 \) cm per hour. The equation is \( y = 28.2-0.1(x - 8) \). For \( x = 5 \): \( y=28.2-0.1\times(5 - 8)=28.2 + 0.3 = 28.5 \). But the button is 5 and x. Alternatively, maybe the problem is to find the height after 10 hours, but the button is 5. Wait, maybe I made a mistake in the problem understanding.

Wait, let's start over.

Let the linear function be \( h(t)=mt + b \), where \( h(t) \) is the height at time \( t \) (in hours).

We know that \( h(8)=28.2 \) and \( h(26)=26.4 \).

So we have the system of equations:

\( 8m + b=28.2 \)

\( 26m + b=26.4 \)

Subtract the first equation from the second:

\( (26m + b)-(8m + b)=26.4 - 28.2 \)

\( 18m=-1.8 \)

\( m=-0.1 \)

Then substitute \( m = - 0.1 \) into \( 8m + b=28.2 \):

\( 8\times(-0.1)+b=28.2 \)

\( - 0.8 + b=28.2 \)

\( b=29 \)

So the equation is \( h(t)=-0.1t + 29 \)

Now, if we want to find \( h(10) \):

\( h(10)=-0.1\times10 + 29=28 \)

If we want to find \( h(5) \):

\( h(5)=-0.1\times5 + 29=28.5 \)

But the button in the image is 5 and x, and the text box is for centimeters. Maybe the problem is to find the height after 5 hours?

Wait, maybe the original problem was "After 10 hours"? No, the button is 5. Wait, perhaps there was a mis - typing. But according to the calculation, if \( t = 10 \), \( h = 28 \); if \( t = 5 \), \( h = 28.5 \).

But let's check the problem statement again. The user's image: "After 8 hours of burning, a candle has a height of 28.2 centimeters. After 26 hours of burning, its height is 26.4 centimeters. What is the height of the candle after 10 hours?" Wait, maybe the "10" is a typo, and it's "5"?

Alternatively, maybe I made a mistake. Let's calculate the rate of change again. The change in height is \( 26.4 - 28.2=-1.8 \) cm over \( 26 - 8 = 18 \) hours. So the rate is \( -1.8\div18=-0.1 \) cm per hour. So every hour, the candle loses 0.1 cm in height.

From 8 hours to 10 hours: 2 hours pass. So the height decreases by \( 0.1\times2 = 0.2 \) cm. So the height at 10 hours is \( 28.2-0.2 = 28 \) cm.

Answer:

28 (If the time is 10 hours) or 28.5 (If the time is 5 hours. But based on the button with 5, maybe the time is 5 hours. Wait, maybe the problem was "After 10 hours" is a mis - print, and it's "After 5 hours". But according to the calculation for \( t = 10 \), the answer is 28. If we consider the button with 5, maybe the intended time is 5, and the answer is 28.5. But let's go with the problem as stated: "after 10 hours". So the answer is 28.