# Tutor profile: Israel M.

## Questions

### Subject: SAT II Mathematics Level 2

Baron deposits $400 into a new savings account that gains 2.4% compound interest every 3 months. How many years will it take for his savings account to grow large enough to support his future son's $60,000 college tuition?

Baron starts off with $400 dollars. This increases to $400 + ($400 * 2.4)/100 with the added compound interest after the 3rd month. This can be rewritten as $400 (1 + 0.024). Since we increase the amount every 3 months, we can find the total savings by multiplying the number of years by 4 to factor in the change that happens every 3 months. Now, the expression to find the number of years it takes to reach a savings of $60,000 can be written as $$ 60,000 = 400*(1+0.024)^{4x} $$ for x equals the number of years the money has been sitting since the first $400 were deposited. Finally, we can solve for x by first dividing both sides by 400 which gives us $$ 1,500= (1+0.024)^{4x} $$. Then, we take both sides to the power of (1/4) to give us $$ 6.22= (1+0.024)^{x} $$. Next, we take the log of both sides to the base of 1.024 to give us $$ log_{1.024}(6.22)= x $$. Solving for x, we find it takes 77. 09 years (just over 77 years) for Baron to earn enough for his son's tuition.

### Subject: Economics

In a busy New York suburb, if a ice-cream parlor with 12 employees and 3 soft serve machines can serve 150 cones of ice-cream a day, is the company experiencing increasing, decreasing, or constant returns to scale if they produce 300 cones of ice-cream after increasing their employment to 23 employees and 5 soft serve machines?

The company is experiencing an increasing return to scale. Since the output was increased by 100% but required an increase in input of less than 100% (since 12 to 23 is about a 92% increase and from 3 to 5 is a 67% increase), we know the rate of increase was more than constant and is thus an increasing returns to scale.

### Subject: Algebra

Determine if the following expression is linear or not. Explain why. y = x^2

The expression is nonlinear because as the input increases, the output increases exponentially. In other words, the rate of increase (or slope) is not constant and is therefor not linear. We can also test this by first looking at a linear model, say y = 2x, and plug in values from 1 to 10. We get a constant rate of increase in the slope by using the slope formula (y2-y1)/(x2-x1) to test each slope. Alternatively, when testing the model of y = x^2, we find a an increasing slope between each point tested from 1 to 10.

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