Candy The following data represents the weight (in grams) of various candy bars and the corresponding number of calories: Candy Bar Weight (x) Calories Hershey’s Milk Chocolate 44.28 230 Nestle’s Crunch 44.84 230 Butterfinger 61.30 270 Baby Ruth 66.45 280 Almond Joy 47.33 220 A. Use the lightest (least weight) and heaviest (greatest weight) candy bars to write the equation of a line that goes through them: B. Use your equation in A to estimate the number of calories in a candy bar that weighs 68.3 g. C. Use technology to calculate the line of best fit using the least squares method. D. Interpret the slope in words.
The Correct Answer and Explanation is:
To answer the problem step-by-step:
A. Equation of a line using the lightest and heaviest candy bars
From the data:
- The lightest candy bar is Almond Joy with a weight of 44.28g and 230 calories.
- The heaviest candy bar is Baby Ruth with a weight of 66.45g and 280 calories.
To write the equation of a line passing through these two points, we need to use the slope-intercept form of a line, which is:
y=mx+by = mx + by=mx+b
Where:
- yyy is the dependent variable (calories),
- xxx is the independent variable (weight),
- mmm is the slope of the line,
- bbb is the y-intercept.
First, we need to calculate the slope (m) using the formula: m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}m=x2−x1y2−y1
Using the points (44.28, 230) and (66.45, 280): m=280−23066.45−44.28=5022.17≈2.26m = \frac{280 – 230}{66.45 – 44.28} = \frac{50}{22.17} \approx 2.26m=66.45−44.28280−230=22.1750≈2.26
Now, we can use one of the points, say (44.28, 230), and substitute into the equation y=mx+by = mx + by=mx+b to solve for bbb: 230=2.26(44.28)+b230 = 2.26(44.28) + b230=2.26(44.28)+b 230=100.04+b230 = 100.04 + b230=100.04+b b=230−100.04=129.96b = 230 – 100.04 = 129.96b=230−100.04=129.96
Thus, the equation of the line is: y=2.26x+129.96y = 2.26x + 129.96y=2.26x+129.96
B. Estimate the number of calories in a candy bar that weighs 68.3g
To estimate the calories for a candy bar that weighs 68.3g, we substitute x=68.3x = 68.3x=68.3 into the equation y=2.26x+129.96y = 2.26x + 129.96y=2.26x+129.96: y=2.26(68.3)+129.96y = 2.26(68.3) + 129.96y=2.26(68.3)+129.96 y=154.06+129.96=284.02y = 154.06 + 129.96 = 284.02y=154.06+129.96=284.02
So, the estimated number of calories is approximately 284 calories.
C. Line of best fit using technology (least squares method)
Using a calculator or software such as Excel or a graphing calculator, you can find the line of best fit using the least squares method. In this case, input the weights and calories and the technology will provide the best fitting line. The line of best fit should be very similar to the one we manually derived, but with more precision.
For instance, in Excel, you can use the “LINEST” function or plot the points and add a trendline to obtain the equation.
D. Interpret the slope in words
The slope of the line is 2.26, which means that for every additional gram of weight, the candy bar contains approximately 2.26 more calories. So, as the weight of the candy bar increases, the number of calories increases at a rate of 2.26 calories per gram of weight.
