Question

suppose that in a memory experiment the rate of memorizing is given by ( m(t) = -0.004t^2 + 0.4t ), where ( m(t) ) is the memory rate, in words per minute. how many words are memorized in the first 13 minutes? (\boxed{\text{words}}) round your answer to the nearest whole word

Explanation:

Step1: Identify the integral

To find the total number of words memorized in the first 13 minutes, we need to integrate the rate function \( M'(t) \) from \( t = 0 \) to \( t = 13 \). So we calculate \( \int_{0}^{13} (-0.004t^{2}+0.4t) dt \).

Step2: Integrate term by term

The integral of \( -0.004t^{2} \) with respect to \( t \) is \( -0.004\times\frac{t^{3}}{3}=-\frac{0.004}{3}t^{3} \), and the integral of \( 0.4t \) with respect to \( t \) is \( 0.4\times\frac{t^{2}}{2} = 0.2t^{2} \). So the antiderivative \( F(t) \) is \( -\frac{0.004}{3}t^{3}+0.2t^{2} \).

Step3: Evaluate the definite integral

We evaluate \( F(13)-F(0) \). First, calculate \( F(13) \):
\( -\frac{0.004}{3}(13)^{3}+0.2(13)^{2} \)
\( (13)^{3}=2197 \), so \( -\frac{0.004}{3}\times2197\approx - \frac{8.788}{3}\approx - 2.929 \)
\( (13)^{2} = 169 \), so \( 0.2\times169 = 33.8 \)
Then \( F(13)\approx - 2.929 + 33.8 = 30.871 \)
\( F(0)=-\frac{0.004}{3}(0)^{3}+0.2(0)^{2}=0 \)
So the integral from 0 to 13 is \( 30.871-0 = 30.871 \)

Step4: Round to nearest whole number

Rounding 30.871 to the nearest whole number gives 31.

Answer:

31