The number of bacteria in a refrigerated food product is given by \( N(T)=23 T^{2}-62 T+24 \), \( 3

Ask by Maxwell Allan. in the United States

Jan 21,2025

To find the composite function \( N(T(t)) \), we first substitute \( T(t) \) into \( N(T) \). 1. Start with the expression for \( N(T) \): \[ N(T) = 23T^2 - 62T + 24 \] 2. Substitute \( T(t) = 6t + 1.9 \): \[ N(T(t)) = N(6t + 1.9) = 23(6t + 1.9)^2 - 62(6t + 1.9) + 24 \] 3. Calculate \( (6t + 1.9)^2 \): \[ (6t + 1.9)^2 = 36t^2 + 2 \cdot 6t \cdot 1.9 + (1.9)^2 = 36t^2 + 22.68t + 3.61 \] 4. Substitute back into \( N(T(t)) \): \[ N(T(t)) = 23(36t^2 + 22.68t + 3.61) - 62(6t + 1.9) + 24 \] \[ = 828t^2 + 521.64t + 83.03 - 372t - 117.8 + 24 \] \[ = 828t^2 + (521.64 - 372)t + (83.03 - 117.8 + 24) \] \[ = 828t^2 + 149.64t - 10.77 \] So, the composite function \( N(T(t)) \) is: \[ N(T(t)) = 828t^2 + 149.64t - 10.77 \] Next, we find the number of bacteria after \( t = 1.4 \) hours: 1. Substitute \( t = 1.4 \) into \( N(T(t)) \): \[ N(T(1.4)) = 828(1.4)^2 + 149.64(1.4) - 10.77 \] 2. Calculate \( 1.4^2 \): \[ 1.4^2 = 1.96 \] 3. Plug values into the function: \[ N(T(1.4)) = 828(1.96) + 149.64(1.4) - 10.77 \] \[ = 1624.08 + 209.496 - 10.77 \] \[ = 1822.806 \] 4. Round to the nearest whole value: \[ \approx 1823 \text{ bacteria} \] So, the number of bacteria after 1.4 hours is approximately **1823 bacteria**.