Alex L.

Tutor for a year, scored a 36 on the ACT

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Basic Chemistry

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Question:

What is the empirical formula for a compound consisting of 35.7% carbon, 33.3% nitrogen, 28.6% oxygen, and 2.4% hydrogen

Alex L.

Answer:

Imagine you are given $$100$$ grams of the unknown compound, This would mean you have: $$35.7$$ grams of carbon $$33.3$$ grams of nitrogen $$28.6$$ grams of oxygen $$2.4$$ grams of hydrogen By dividing the different molar masses of the elements, you can find the number of moles of each: $$ 35.7 g \times \frac{1 mol}{12.01 g } = 2.973 $$ moles of carbon $$ 33.3 g \times \frac{1 mol}{14.01 g } = 2.377 $$ moles of nitrogen $$ 28.6 g \times \frac{1 mol}{16 g } = 1.786 $$ moles of oxygen $$ 2.4 g \times \frac{1 mol}{1.01 g } = 2.376 $$ moles of hydrogen Now take each number of moles and divide by the lowest number of moles. Multiply by 3 to get a ratio of whole numbers: $$ 2.973 / 1.786 = 1.66 \times 3 = 5$$ $$ 2.377 / 1.786 = 1.33 \times 3 = 4$$ $$ 1.786 / 1.786 = 1.00 \times 3 = 3$$ $$ 2.376 / 1.786 = 1.33 \times 3 = 4$$ This ratio, 5:4:3:4, gives you everything necessary to write the empirical formula, $$ C_5N_4O_3H_4 $$

Calculus

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Question:

Find the area under the curve $$ y = 4x^3 $$ from $$ x = -1 $$ to $$ x = 4 $$.

Alex L.

Answer:

Set up the definite integral to find the area on the given interval: $$\int_{-1}^{4}4x^3$$ and solve by increasing the exponent by one and dividing by that number $$ = \big[\frac{4x^{3+1}}{4} \big]_{-1}^{4} = \big[x^4\big]_{-1}^{4} = (4)^4 - (-1)^4 = 256 - 1 = \textbf{255} $$

ACT

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Question:

Square A has a side length of $$ 16 $$ and contains a smaller Square B bounded by the midpoints of Square A's sides. What is the area of the largest circle that could be fit inside Square B? A) $$ 49\pi $$ B) $$ 64\pi $$ C) $$ 32\pi $$ D) $$ 128\pi $$

Alex L.

Answer:

Answer: C) $$ 32\pi $$ The first step is to find the side length of Square B. A $$ 45^{\circ}-45^{\circ}-90^{\circ} $$ triangle can be formed using two legs, each extending from a corner of Square A to a midpoint. The hypotenuse of that triangle forms one side of Square B. Since the side length of Square A is $$ 16 $$, the distance from a corner to a midpoint is half that length, or $$ 8 $$. The hypotenuse of a $$ 45^{\circ}-45^{\circ}-90^{\circ} $$ triangle is equal to the length of one of the legs times the square root of 2, making the length of the hypotenuse of this particular triangle $$ 8\sqrt2 $$, which is the side length of Square B. This side length also represents the diameter of the largest circle that could fit inside Square B. To find the radius of the circle, divide the diameter in half to get $$ 4\sqrt2 $$. Plug this value into the equation for the area of a circle, $$ A = \pi r^2 $$ to obtain the final answer, $$ 32\pi $$.

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