Question

an object travels along a straight line. the function s(t) = 2t³ lnt - t² gives the objects position, in meters, at time t > 0 minutes. what is the objects speed, in meters per minute, at t = 3? write your answer as an integer or decimal rounded to the nearest tenth. meters per minute

Explanation:

Step1: Recall Speed as Derivative of Position

Speed is the magnitude of the velocity, and velocity \( v(t) \) is the derivative of the position function \( s(t) \). So we first find \( s'(t) \). Given \( s(t) = 2t^3 \ln t - t^2 \), we use the product rule and power rule. The product rule is \( (uv)' = u'v + uv' \), where \( u = 2t^3 \), \( v = \ln t \), and the derivative of \( -t^2 \) is \( -2t \).

First, find \( u' \): \( u = 2t^3 \), so \( u' = 6t^2 \). \( v = \ln t \), so \( v' = \frac{1}{t} \). Then by product rule, \( (2t^3 \ln t)' = 6t^2 \ln t + 2t^3 \cdot \frac{1}{t} = 6t^2 \ln t + 2t^2 \). Then add the derivative of \( -t^2 \), so \( s'(t) = 6t^2 \ln t + 2t^2 - 2t \).

Step2: Evaluate Derivative at \( t = 3 \)

Now substitute \( t = 3 \) into \( s'(t) \). First, calculate each term:

- \( 6t^2 \ln t \) at \( t = 3 \): \( 6 \times 3^2 \times \ln 3 = 6 \times 9 \times \ln 3 = 54 \ln 3 \). \( \ln 3 \approx 1.0986 \), so \( 54 \times 1.0986 \approx 54 \times 1.1 = 59.4 \) (more accurately, \( 54 \times 1.0986 = 59.3244 \)).
- \( 2t^2 \) at \( t = 3 \): \( 2 \times 3^2 = 2 \times 9 = 18 \).
- \( -2t \) at \( t = 3 \): \( -2 \times 3 = -6 \).

Now sum these terms: \( 59.3244 + 18 - 6 = 59.3244 + 12 = 71.3244 \). Rounded to the nearest tenth, it's \( 71.3 \). Wait, wait, maybe I miscalculated \( 54 \ln 3 \). Let's recalculate \( \ln 3 \approx 1.098612 \), so \( 54 \times 1.098612 = 54 \times 1.098612 = 59.325048 \). Then \( 59.325048 + 18 = 77.325048 - 6 = 71.325048 \), which is approximately \( 71.3 \) when rounded to the nearest tenth? Wait, no, 71.325 is 71.3 when rounded to the nearest tenth? Wait, 0.325, the hundredth digit is 2, so tenths place is 3, and hundredth is 2, so 71.3? Wait, no, 71.325, the tenths digit is 3, hundredth is 2, so we look at the thousandth, which is 5? Wait, no, 59.3244 + 18 is 77.3244, then minus 6 is 71.3244. So 71.3244, rounded to the nearest tenth: the tenths digit is 3, hundredths is 2, which is less than 5, so it's 71.3? Wait, but maybe I made a mistake in the derivative. Wait, let's recheck the derivative.

Wait, original function: \( s(t) = 2t^3 \ln t - t^2 \). Derivative of \( 2t^3 \ln t \): using product rule, \( u = 2t^3 \), \( u' = 6t^2 \); \( v = \ln t \), \( v' = 1/t \). So \( (uv)' = u'v + uv' = 6t^2 \ln t + 2t^3 \times (1/t) = 6t^2 \ln t + 2t^2 \). Then derivative of \( -t^2 \) is \( -2t \). So \( s'(t) = 6t^2 \ln t + 2t^2 - 2t \). That's correct.

Now plug \( t = 3 \):

\( 6*(3)^2*\ln(3) + 2*(3)^2 - 2*(3) \)

\( 6*9*\ln(3) + 2*9 - 6 \)

\( 54\ln(3) + 18 - 6 \)

\( 54\ln(3) + 12 \)

\( \ln(3) \approx 1.098612 \), so \( 54*1.098612 = 59.325048 \)

Then \( 59.325048 + 12 = 71.325048 \), which is approximately 71.3 when rounded to the nearest tenth. Wait, but maybe the problem has a typo? Wait, the original function: is it \( 2t^3 \ln t - t^2 \)? Or maybe \( 2t^3 \ln t - t^2 \) is correct. Alternatively, maybe I made a mistake in the derivative. Wait, no, product rule is correct. Let's check with another approach. Let's compute \( s'(t) \) again:

\( d/dt [2t^3 \ln t] = 2 [3t^2 \ln t + t^3 (1/t)] = 2[3t^2 \ln t + t^2] = 6t^2 \ln t + 2t^2 \). Then \( d/dt [-t^2] = -2t \). So \( s'(t) = 6t^2 \ln t + 2t^2 - 2t \). Correct.

So at \( t = 3 \), \( 6*(9)*\ln(3) + 2*(9) - 2*(3) = 54\ln(3) + 18 - 6 = 54\ln(3) + 12 \). \( \ln(3) \approx 1.0986 \), so 54*1.0986 = 59.3244, plus 12 is 71.3244, which is 71.3 when rounded to the nearest tenth. Wait, but maybe the problem is \( s(t) = 2t^3 \ln t - t^2 \), or maybe \( \ln t \) is \( \ln(t) \), yes. So the speed at \( t = 3 \) is approximately 71.3 meters per minute. Wait, but let's check with calculator:

Calculate \( 54\ln(3) \):

\( \ln(3) \approx 1.098612289 \)

\( 54 * 1.098612289 = 54 * 1.098612289 \)

54 * 1 = 54, 54 * 0.098612289 = 54 * 0.1 = 5.4, minus 54 * 0.001387711 = 0.0749364, so 5.4 - 0.0749364 = 5.3250636. So total 54 + 5.3250636 = 59.3250636. Then add 12: 59.3250636 + 12 = 71.3250636, which is 71.3 when rounded to the nearest tenth.

Answer:

71.3