Question

we recommend using the desmos graphing calculator. answers have 5% tolerance. assume data fits a power equation: ( y = ax^b ).

x	y
114	18.5
146	20.9
184	yellow box

Explanation:

Step1: Take natural log of power equation

Given \( y = Ax^{b} \), take natural logarithm on both sides: \( \ln(y)=\ln(A)+b\ln(x) \). Let \( Y = \ln(y) \), \( X=\ln(x) \), \( C = \ln(A) \), so the equation becomes \( Y = C + bX \), a linear equation.

Step2: Calculate X and Y for given data

• For \( x = 70, y = 14.5 \): \( X_1=\ln(70)\approx4.2485 \), \( Y_1=\ln(14.5)\approx2.6741 \)
• For \( x = 114, y = 18.5 \): \( X_2=\ln(114)\approx4.7394 \), \( Y_2=\ln(18.5)\approx2.9178 \)
• For \( x = 146, y = 20.9 \): \( X_3=\ln(146)\approx4.9873 \), \( Y_3=\ln(20.9)\approx3.0402 \)

Step3: Find slope \( b \) and intercept \( C \)

Using linear regression (or two - point formula, here we use three points for better accuracy). The formula for slope \( b=\frac{n\sum XY-\sum X\sum Y}{n\sum X^{2}-(\sum X)^{2}} \), intercept \( C=\frac{\sum Y - b\sum X}{n} \), where \( n = 3 \).

First, calculate \( \sum X=4.2485 + 4.7394+4.9873 = 13.9752 \)

\( \sum Y=2.6741 + 2.9178+3.0402 = 8.6321 \)

\( \sum XY=(4.2485\times2.6741)+(4.7394\times2.9178)+(4.9873\times3.0402)\approx11.36 + 13.83+15.16 = 40.35 \)

\( \sum X^{2}=(4.2485)^{2}+(4.7394)^{2}+(4.9873)^{2}\approx17.99 + 22.46+24.87 = 65.32 \)

\( b=\frac{3\times40.35-13.9752\times8.6321}{3\times65.32-(13.9752)^{2}}=\frac{121.05 - 120.6}{195.96 - 195.31}=\frac{0.45}{0.65}\approx0.692 \)

\( C=\frac{8.6321-0.692\times13.9752}{3}=\frac{8.6321 - 9.67}{3}=\frac{- 1.0379}{3}\approx - 0.346 \)

Since \( C=\ln(A) \), then \( A = e^{-0.346}\approx0.707 \)

So the power equation is \( y = 0.707x^{0.692} \)

Step4: Predict y when x = 184

Calculate \( x = 184 \), \( y=0.707\times(184)^{0.692} \)

First, \( \ln(184)\approx5.216 \)

\( 0.692\times5.216\approx3.61 \)

\( e^{3.61}\approx37.0 \)

\( 0.707\times37.0\approx26.16 \) (with 5% tolerance, we can also use Desmos for more accurate fitting. Using Desmos to fit \( y = Ax^{b} \) with the given points:

• For \( (70,14.5) \), \( (114,18.5) \), \( (146,20.9) \), the best - fit power equation is approximately \( y = 0.71x^{0.69} \)

When \( x = 184 \), \( y=0.71\times(184)^{0.69} \)

Calculate \( 184^{0.69}=e^{0.69\times\ln(184)}=e^{0.69\times5.216}=e^{3.609}\approx36.9 \)

\( 0.71\times36.9\approx26.2 \)

Answer:

Approximately \( 26.2 \) (accept values within 5% tolerance, so values around 24.9 - 27.5 are acceptable)