If S = (4t + 3)(t - 2), find ds/dt when t = 5 secs.

1. Analyze the question

We are given a function $S$ in terms of $t$, defined as $S = (4t + 3)(t - 2)$. We need to find the derivative of $S$ with respect to $t$ ($\frac{dS}{dt}$) and then evaluate that derivative at $t = 5$.

2. Explain the concept

To solve this, we use the Product Rule of differentiation or expand the brackets first. The expansion method is often simpler for polynomials.

• Expansion: Multiply the two binomials to get a standard polynomial.
• Power Rule: If $y = ax^n$, then $\frac{dy}{dx} = n \cdot ax^{n-1}$.
• Differentiation: Once expanded, we differentiate term by term.

3. Show the solution

Step 1: Expand the expression for $S$ $S = (4t + 3)(t - 2)$ $S = 4t(t) + 4t(-2) + 3(t) + 3(-2)$ $S = 4t^2 - 8t + 3t - 6$ $S = 4t^2 - 5t - 6$

Step 2: Differentiate $S$ with respect to $t$ Using the power rule: $\frac{dS}{dt} = \frac{d}{dt}(4t^2) - \frac{d}{dt}(5t) - \frac{d}{dt}(6)$ $\frac{dS}{dt} = (2 \times 4)t^{2-1} - (1 \times 5)t^{1-1} - 0$ $\frac{dS}{dt} = 8t - 5$

Step 3: Evaluate at $t = 5$ Substitute $t = 5$ into the derivative: $\frac{dS}{dt} = 8(5) - 5$ $\frac{dS}{dt} = 40 - 5$ $\frac{dS}{dt} = 35$

4. Identify the answer

The result is 35 units per sec. Comparing this to the given options: A. 50 B. 35 C. 22 D. 13

The correct option is B.

5. Tip

When you see a product of two brackets, always check if expanding them is faster than using the Product Rule formula $[u \frac{dv}{dt} + v \frac{du}{dt}]$. For simple polynomials like this, expansion saves time and reduces the risk of errors during your JAMB exam!