g) Estimate sin29^circ by linear approximation (3 Marks) h) A monopolist faces the demand function given by x=10-0.5p

g) To estimate $\sin(29^\circ)$ using linear approximation, we can use the formula for linear approximation:<br /><br />$\sin(x) \approx \sin(a) + \cos(a)(x - a)$<br /><br />where $a$ is a value close to $29^\circ$ for which we know the sine and cosine values. A common choice is $a = 30^\circ$, since $30^\circ$ is a well-known angle with known sine and cosine values.<br /><br />$\sin(30^\circ) = \frac{1}{2}$ and $\cos(30^\circ) = \frac{\sqrt{3}}{2}$<br /><br />Now, we can substitute these values into the linear approximation formula:<br /><br />$\sin(29^\circ) \approx \sin(30^\circ) + \cos(30^\circ)(29^\circ - 30^\circ)$<br /><br />$\sin(29^\circ) \approx \frac{1}{2} + \frac{\sqrt{3}}{2}(-1^\circ)$<br /><br />Using a calculator, we can evaluate this expression:<br /><br />$\sin(29^\circ) \approx \frac{1}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\pi}{180}$<br /><br />$\sin(29^\circ) \approx 0.4839$<br /><br />Therefore, the estimated value of $\sin(29^\circ)$ using linear approximation is approximately $0.4839$.<br /><br />h) To find the revenue function for the monopolist, we need to first find the price function $p(x)$ using the given demand function $x = 10 - 0.5p$.<br /><br />Rearranging the demand function, we have:<br /><br />$p = 20 - 2x$<br /><br />Now, we can find the revenue function $R(x)$ by multiplying the price function $p(x)$ by the quantity $x$:<br /><br />$R(x) = p(x) \cdot x$<br /><br />Substituting the price function, we have:<br /><br />$R(x) = (20 - 2x) \cdot x$<br /><br />Simplifying, we get:<br /><br />$R(x) = 20x - 2x^2$<br /><br />Therefore, the revenue function for the monopolist is $R(x) = 20x - 2x^2$.