# Tutor profile: Anna S.

## Questions

### Subject: Calculus

Evaluate the antiderivative of the function $$f(x) = 3x^2 + 2$$ on the interval [0,2].

Using integration rules, we have $$F(x) = \int{f(x)}dx = \frac{3x^3}{3} + 2x,$$ which simplifies to $$F(x) = x^3 + 2x.$$ We can now evaluate using the Fundamental Theorem of Calculus, which states that $$\int_0^2{f(x)}dx = F(2) - F(0).$$ This gives $$\int_0^2{(3x^2 + 2)}dx = 2^3 + 2(2) - [0^3 + 2(0)] = 8 + 4 = 12.$$

### Subject: Algebra

A local pizzeria charges $12 for a large pie and $8 for a small pie. You spent a total of $48 and purchased a total of 5 pizzas. How many large and small pies did you order?

We can solve this problem using a system of equations. Let's represent the number of small pies with the variable S and the number of large pies with the variable L. Since small pies are $8 each, the cost of small pies is 8S. Similarly, since large pies are $12 each, the total cost of large pies ordered is 12L. Therefore, the total cost is 8S + 12L = 48. We also know that 5 pies were ordered in total, so our second equation is S + L = 5. We have two equations and two unknowns, so we are ready to solve for S and L. S + L = 5 S = 5 - L [solve for S] Using substitution: 8S + 12L = 48 8(5 - L) + 12L = 48 [substitute expression for S] 40 - 8L + 12L = 48 [distribute] 40 + 4L = 48 [combine like terms] 4L = 48 - 40 4L = 8 L = 2 Using substitution: S + L = 5 S + 2 = 5 [substitute value for L] S = 3 Solution: L = 2, S = 3 You ordered 2 large pies and 3 small pies!

### Subject: Physics

When a force of 10N is applied to a ball, it accelerates at a rate of 5 m/s$${^2}.$$ The same ball is attached to a string of length 3 m and swung in a circular motion, creating a tension of 24 N in the string. What is the ball's velocity?

The tension on the string can be represented by the equation $$T = \frac{mv^2}{r}$$, where m is mass, v is velocity, and r is the string's radius. We have that T = 24 N and r = 3 m. To solve for velocity, we need to know the ball's mass. To find the mass, we can plug initial values for force and acceleration into the equation $$F = ma.$$ This gives 10 = m(5), so m = 2 kg. Finally, we can solve for velocity. 24 = $$\frac{2 v^2}{3}$$ $$\rightarrow$$ 72 = 2v$$^2$$ $$\rightarrow$$ v$$^2$$ = 36 $$\rightarrow$$ v = 6 m/s. The ball's velocity is 6 m/s.