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Tutor profile: Josh O.

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Josh O.
Experienced Math Tutor and Data Science Enthusiast
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Questions

Subject: Pre-Calculus

TutorMe
Question:

What is the minimum period of $$cos(sin(\theta))$$?

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Josh O.
Answer:

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 $$\pi$$.

Subject: Calculus

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Question:

Given the equation of motion for a rocket is $(h(t) = -16t^2 + 400t + 200,$) what is the velocity after $$5$$ seconds?

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Josh O.
Answer:

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.

Subject: Algebra

TutorMe
Question:

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?

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Josh O.
Answer:

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.

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