A campfire has a flame 24 inches tall after burning for 4 hours. After 6 hours, it is 20 inches tall. Write a linear equation to model the relationship between height h of the fire and time t. Predict how tall the fire will be after burning 8 hours.
show work:_____
height of fire after 8 hours?:_______
You have two points: (4,24) and (6,20)
The slope of that line is -2
so the equation in point-slope form is
y-24 = -2(x-4)
now plug in x=8
or, look at the data to see that it burned 4 inches in two hours, so it will burn another 4 inches in the next two hours.
To write a linear equation to model the relationship between the height h of the fire and time t, we can use the point-slope form of a linear equation.
Step 1: Determine two points on the line.
Given that after 4 hours, the fire is 24 inches tall, we have the point (4, 24).
Given that after 6 hours, the fire is 20 inches tall, we have the point (6, 20).
Step 2: Find the slope.
The slope (m) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (4, 24) and (6, 20), we can calculate the slope:
m = (20 - 24) / (6 - 4) = -2
Step 3: Use the slope-intercept form to write the linear equation.
The slope-intercept form of a linear equation is:
y = mx + b
where m is the slope and b is the y-intercept.
Using the slope m from Step 2 and one of the points (4, 24), we can substitute these values into the equation to solve for b:
24 = -2 * 4 + b
24 = -8 + b
b = 32
Therefore, the linear equation that models the relationship between the height h of the fire and time t is:
h = -2t + 32
To predict the height of the fire after burning 8 hours, we substitute t = 8 into the equation:
h = -2(8) + 32
h = -16 + 32
h = 16
So, the height of the fire after burning 8 hours is predicted to be 16 inches.
To write a linear equation to model the relationship between the height of the fire and time, we can use the slope-intercept form of a linear equation, which is represented as y = mx + b, where y is the dependent variable (height), x is the independent variable (time), m is the slope, and b is the y-intercept.
Let's set the height of the fire (h) as the dependent variable and time (t) as the independent variable. We are given two points on the line: (4, 24) and (6, 20).
First, we can calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (20 - 24) / (6 - 4)
m = -4 / 2
m = -2
Next, we need to find the y-intercept (b). We can choose any given point and substitute its coordinates into the equation y = mx + b. Let's use the point (4, 24):
24 = -2 * 4 + b
24 = -8 + b
b = 32
Now we have the slope (m = -2) and the y-intercept (b = 32), so we can write the linear equation:
h = -2t + 32
To predict the height of the fire after 8 hours, substitute t = 8 into the equation:
h = -2(8) + 32
h = -16 + 32
h = 16
Therefore, the fire is predicted to be 16 inches tall after burning for 8 hours.