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A candle is 18 in. tall after burning for 2 hours. After 5 hours, it is 12 in. tall. Write a linear equation to model the relationship between height h of the candle and time t. Predict how tall the candle will be after burning for 8 hours.
To write a linear equation, we can use the point-slope form:
(y-y1) = m(x-x1),
where m is the slope and (x1, y1) is a point on the line.
Let's use the information given to find the equation:
Point 1: (2, 18) - after burning for 2 hours, the candle height is 18 in.
Point 2: (5, 12) - after burning for 5 hours, the candle height is 12 in.
First, let's find the slope (m):
m = (y2 - y1)/(x2 - x1)
m = (12 - 18)/(5 - 2)
m = -6/3
m = -2
Using point 1, (2, 18), we can substitute the values into the point-slope equation:
(y - 18) = -2(x - 2)
Next, let's solve for y:
y - 18 = -2x + 4
Adding 18 to both sides:
y = -2x + 22
Therefore, the equation that models the relationship between the height h of the candle and the time t is:
h = -2t + 22.
To predict how tall the candle will be after burning for 8 hours, we can substitute in t = 8 into the equation:
h = -2(8) + 22
h = -16 + 22
h = 6.
Therefore, the candle will be 6 inches tall after burning for 8 hours.
To find a linear equation that models the relationship between the height h of the candle and the time t, we can use the slope-intercept form of a linear equation, which is y = mx + b.
Let's assign h as the dependent variable (y) and t as the independent variable (x). We have two points on the line: (2, 18) and (5, 12).
Step 1: Find the slope (m)
Using the formula for slope: m = (y2 - y1) / (x2 - x1)
m = (12 - 18) / (5 - 2)
m = -6 / 3
m = -2
Step 2: Substitute one point and the slope into the slope-intercept form
Using the point-slope form: y - y1 = m(x - x1)
Using the point (2, 18) and the slope -2:
y - 18 = -2(x - 2)
Step 3: Convert to slope-intercept form
Distribute -2 on the right side of the equation:
y - 18 = -2x + 4
Move -18 to the right side:
y = -2x + 22
The linear equation that models the relationship between the height h of the candle and the time t is h = -2t + 22.
To predict how tall the candle will be after burning for 8 hours, substitute t = 8 into the equation:
h = -2(8) + 22
h = -16 + 22
h = 6
Therefore, the predicted height of the candle after burning for 8 hours will be 6 inches.