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.
Let's assume that the linear equation to model the relationship between the height of the fire, h, and the time, t, is of the form h = mt + b, where m is the slope of the line and b is the y-intercept.
To find the slope, we can use the formula m = (y2 - y1) / (x2 - x1) with the points (4, 24) and (6, 20).
m = (20 - 24) / (6 - 4)
m = -4 / 2
m = -2
Using the point-slope form of a linear equation, we can now substitute one of the points, (4, 24), into the equation to find b:
24 = -2(4) + b
24 = -8 + b
b = 24 + 8
b = 32
Therefore, the linear equation that models the relationship between the height of the fire, h, and the time, t, is h = -2t + 32.
To predict how tall the fire will be after burning 8 hours, we substitute t = 8 into the equation:
h = -2(8) + 32
h = -16 + 32
h = 16
Therefore, the fire will be 16 inches tall after burning for 8 hours.
To write a linear equation to model the relationship between the height h of the fire and time t, we need to find the slope and the y-intercept.
Using the two data points given, we can calculate the slope (m) using the formula:
m = (h2 - h1) / (t2 - t1)
Let's use (h1, t1) = (24 inches, 4 hours) and (h2, t2) = (20 inches, 6 hours):
m = (20 - 24) / (6 - 4)
m = -4 / 2
m = -2
Next, we need to find the y-intercept (b). We can use one of the data points and the slope in the equation:
h = mt + b
Using (h, t) = (24 inches, 4 hours):
24 = -2(4) + b
24 = -8 + b
b = 32
Therefore, the linear equation is:
h = -2t + 32
To predict the height of the fire after burning 8 hours, substitute t = 8 into the equation:
h = -2(8) + 32
h = -16 + 32
h = 16 inches
Therefore, the fire will be 16 inches tall after burning for 8 hours.
To write a linear equation that models the relationship between the height of the fire (h) and the time it has been burning (t), we can use the slope-intercept form of a linear equation, which is:
h = mx + b
Where:
- h is the height of the fire,
- t is the time the fire has been burning,
- m is the slope of the line (representing the rate of change in height), and
- b is the y-intercept (representing the initial height of the fire).
To find the values of m and b, we can use the given information: after burning for 4 hours, the fire has a height of 24 inches, and after 6 hours, it has a height of 20 inches.
Using the formula for calculating the slope (m) of a line:
m = (change in height) / (change in time)
Substituting the given values:
m = (20 - 24) / (6 - 4)
= -4 / 2
= -2
So, the slope (m) of the linear equation is -2.
Now, to find the value of the y-intercept (b), we can substitute the values of h and t from one of the given points. Let's use the point (4, 24):
24 = -2(4) + b
24 = -8 + b
Adding 8 to both sides:
b = 32
Therefore, the linear equation that models the relationship between the height of the fire (h) and the time it has been burning (t) is:
h = -2t + 32
To predict the height of the fire after burning for 8 hours, we can substitute t = 8 into the equation:
h = -2(8) + 32
h = -16 + 32
h = 16
Therefore, the predicted height of the fire after burning for 8 hours is 16 inches.