#### To determine

**To calculate:** The rate of change of the length of Alaskan rockfish at age 12.

#### Answer

The instantaneous rate of change of the length of Alaskan rockfish at age 12 is 1.718 inches per year.

#### Explanation

**Given:**

The length (*L*) of Alaskan rockfish at age *A* is L=0.0155A3−0.372A2+3.95A+1.21 where *L* is measured in inches and *A* in years.

**Derivative rules:**

(1) Derivative of Constant Function: ddt(c)=0

(2) Constant Multiple Rule: ddx[c⋅f(x)]=c⋅ddxf(x)

(3) Power Rule: ddx(xn)=nxn−1

(4) Difference Rule: ddx[f(x)−g(x)]=ddx(f(x))−ddx(g(x))

(5) Sum Rule: ddx[f(x)+g(x)]=ddx(f(x))+ddx(g(x))

**Calculation:**

Obtain the first derivative of *L* with respect to *A.*

dLdA=ddA(0.0155A3−0.372A2+3.95A+1.21)

Apply the derivative rules (4) and (5) to get,

dLdA=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)+ddA(1.21)

Apply the derivative rules (1) and (2) to get

dLdA=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)+0=ddA(0.0155A3)−ddA(0.372A2)+ddA(3.95A)=0.0155ddA(A3)−0.372ddA(A2)+3.95ddA(A)

Apply the power rule (3) and simplify the terms,

dLdA=0.0155(3A3−1)−0.372(2A2−1)+3.95(1A1−1)+0=0.0155(3A2)−0.372(2A)+3.95(1)=0.0465A2−0.744A+3.95

Here, the derivative of *L* with respect to *A* (dLdA) means the instantaneous rate of change of the length of Alaskan rockfish.

Obtain the values of dLdA when A=12 .

dLdA|A=12=0.0465(12)2−0.744(12)+3.95=0.0465(144)−0.744(12)+3.95=6.696−8.928+3.95=1.718

Therefore, the instantaneous rate of change of the length of Alaskan rockfish at age 12 is 1.718 inches per year.