Tutor profile: Josh O.
What is the minimum period of $$cos(sin(\theta))$$?
We know the period of $$sin(\theta)$$ and $$\cos(\theta)$$ individually, which is $$2\pi$$. However, all we need to do is examine values as we vary the angle in the $$\sin$$ function, which is the argument to the $$\cos$$ function. The $$cos$$ function is even, meaning that $$cos(-\theta) = cos(\theta)$$, so as $$\theta$$ goes between $$0$$ and $$\pi$$, $$sin(\theta)$$ goes between $$0$$ and $$1$$, then to $$0$$ and back down to $$-1$$ over $$\pi$$ to $$2\pi$$. But, $$cos$$ gives the same outputs when the argument is negative or positive, so it gives the same outputs as $$\theta$$ goes from $$0$$ to $$\pi$$, then $$\pi$$ to $$2\pi$$. Therefore, the period is $$\pi$$.
Given the equation of motion for a rocket is $(h(t) = -16t^2 + 400t + 200,$) what is the velocity after $$5$$ seconds?
The velocity equation is given by the first derivative of $$h(t)$$, which is: $(v(t) = -32t+400$). Therefore, $$v(5) = -32(5)+400 = -160+400 = 240$$ m/s.
A jar contains $$20$$ coins. You empty the jar at the bank and the amount in the jar is $$\$ 1.20$$. If there were only nickels and dimes in the jar, how many dimes were there?
We know there are only nickels and dimes in the jar, and we know that nickels are worth $$5$$ cents and dimes are worth $$10$$ cents. Let's call the number of nickels $$N$$ and the number of dimes $$D$$. Since the total amount is $$\$1.20$$, convert this to cents which is $$1.20 \cdot 100 = 120$$ cents. Therefore, we know: $(5N + 10D = 120$). We also know that the number of nickels and dimes must total to the $$20$$ coins in the jar, so we have the system: $(5N+10D = 120 \\ N + D = 20.$) Divide the first equation by $$5$$ to simplify it, resulting in the system: $(N + 2D = 60 \\ N +D = 20.$) We can subtract the second equation from the first (using the "elimination" technique) which will result in: $(D = 40.$) Therefore there are $$40$$ total dimes.