Question

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

x	y
30	87.5
43	105
55	118
74	yellow box

Explanation:

Step1: Take natural log of power equation

Given \( y = Ax^p \), take natural logarithm on both sides: \( \ln y=\ln A + p\ln x \). Let \( Y = \ln y \), \( X=\ln x \), \( B=\ln A \), so the equation becomes \( Y = B + pX \), a linear equation.

Step2: Calculate X and Y for given data

• For \( x = 30, y = 87.5 \): \( X_1=\ln 30\approx3.4012 \), \( Y_1=\ln 87.5\approx4.4716 \)
• For \( x = 43, y = 105 \): \( X_2=\ln 43\approx3.7612 \), \( Y_2=\ln 105\approx4.6539 \)
• For \( x = 55, y = 118 \): \( X_3=\ln 55\approx4.0073 \), \( Y_3=\ln 118\approx4.7707 \)

Step3: Find slope \( p \) and intercept \( B \)

Using linear regression (or two - point formula for approximation). Let's use first and third points:
Slope \( p=\frac{Y_3 - Y_1}{X_3 - X_1}=\frac{4.7707 - 4.4716}{4.0073 - 3.4012}=\frac{0.2991}{0.6061}\approx0.4935 \)
Intercept \( B = Y_1 - pX_1=4.4716-0.4935\times3.4012\approx4.4716 - 1.6785 = 2.7931 \)
Then \( A = e^B=e^{2.7931}\approx16.21 \)

So the power equation is \( y = 16.21x^{0.4935} \)

Step4: Calculate y when x = 74

First, calculate \( \ln 74\approx4.3041 \)
Then \( \ln y=\ln(16.21)+0.4935\times4.3041\approx2.793 + 2.124\approx4.917 \)
Then \( y = e^{4.917}\approx135.4 \) (with 5% tolerance, the range is \( 135.4\times(1 - 0.05)=128.63 \) to \( 135.4\times(1 + 0.05)=142.17 \))

Answer:

Approximately \( 135 \) (or within the range 129 - 142 due to 5% tolerance)