Sshow all the steps that you use to solve this problem in the space provided.
A candle is 17 in. tall after burning for 3 hours. After 5 hours, it is 15 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 8 hours.
For begining mark:
x = the time in hours
y = the height in inches
You know:
A candle is 17 in tall after burning for 3 hours.
After 5 hours, it is 15 in tall.
This mean x1 = 3h , y1 = 17in , x2 = 5h , y2 = 15 in
Find the slope m of the linear equation between this two points.
m = ( y2 - y1 ) / ( x2 - x1 )
substitutes the values
m = ( 15 - 17 ) / ( 5 - 3 ) = - 2 / 2 = - 1
The equation of the line in the point-slope form is:
y - y1 = m ∙ ( x - x1 )
so
y - 17 = ( - 1 ) ∙ ( x - 3 )
y - 17 = - x + 3
Add 17 to both sides
y - 17 + 17 = - x + 3 + 17
y = - x + 20
Replace y with h and x with t
h = - t + 20
For predicting put t = 8 into you equation
h = - t + 20 = - 8 + 20 = 12
After burning 8 hours tall the candle will be 12 in
To model the relationship between the height of the candle and time, we can use the information given and form a linear equation in the form of y = mx + b.
Step 1: Identify the given information:
- The candle is 17 in. tall after burning for 3 hours, which can be represented as the point (3, 17).
- After burning for 5 hours, the candle is 15 in. tall, represented as the point (5, 15).
Step 2: Find the slope:
Slope (m) represents the rate of change of the height of the candle with respect to time. We can calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (3, 17) and (5, 15), we have:
m = (15 - 17) / (5 - 3)
m = -2 / 2
m = -1
Step 3: Write the equation using the slope-intercept form:
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. We can substitute the slope value into the equation.
Using the point (3, 17):
17 = -1(3) + b
17 = -3 + b
b = 20
Step 4: Write the final equation:
After finding the value of the y-intercept, we can substitute the values into the equation.
The linear equation representing the relationship between the height (h) of the candle and time (t) is:
h = -t + 20
Now, we can predict the height of the candle after burning for 8 hours by substituting t = 8 into the equation.
h = -(8) + 20
h = -8 + 20
h = 12
Therefore, the candle is predicted to be 12 inches tall after burning for 8 hours.
To solve this problem, we need to model the relationship between the height of the candle and the time it has been burning using a linear equation. Here are the steps to solve the problem and find the solution:
Step 1: Identify the variables:
- Let's use "h" to represent the height of the candle, and "t" to represent the time it has been burning.
Step 2: Determine the rate of change:
- We are given two data points: after 3 hours, the height is 17 inches, and after 5 hours, the height is 15 inches.
- To find the rate of change, we calculate the difference in height over the difference in time.
- The change in height is: 15 - 17 = -2 inches.
- The change in time is: 5 - 3 = 2 hours.
- So, the rate of change is: -2 inches per 2 hours.
Step 3: Write the linear equation:
- The linear equation will be in the form: h = mt + b, where "m" is the rate of change and "b" is the y-intercept.
- Substituting the known values, we have: h = -2t + b.
- To find "b," we substitute the values of either data point into the equation and solve for "b."
- Let's use the data point (t=3, h=17):
- Substituting those values, we have: 17 = -2(3) + b.
- Solving for "b," we get: b = 17 + 6 = 23.
- So, the linear equation is: h = -2t + 23.
Step 4: Predict the height after 8 hours:
- Substitute the value "t=8" into the equation to find the predicted height:
- Substituting t=8 into the equation, we have: h = -2(8) + 23.
- Solving for "h," we get: h = -16 + 23 = 7 inches.
Therefore, the predicted height of the candle after burning for 8 hours is 7 inches.