# Tutor profile: Kailey C.

## Questions

### Subject: Chemistry

Consider a sample of calcium carbonate $$(CaCO_3)$$ in the form of a cube measuring 2.005 inches on each side. If the sample has a density of $$2.71 g/cm^3$$, how many oxygen atoms does it contain?

We can begin by finding the mass of the cube. Since density is provided in cubic centimeters, we must find the volume in these units. The volume of the cube in inches is $$2.005^3 in ^3$$, or $$8.06015$$ $$in^3$$. We know that $$1 in =2.54cm$$. Therefore, we can find that $$1^3 in^3 = 2.54^3 cm^3 = 16.387 cm^3$$. We multiply this by the volume of the cube in cubic inches to reach the volume in $$cm^3$$ and can then multiply by the density to find the mass. We find that $$8.06015 in^3 \cdot \frac{16.387 cm^3}{1 in^3}\cdot \frac{2.71 g}{1 cm^3} = 357.943 g$$. Next, we must find the molar mass of $$CaCo_3$$. We add the atomic mass of each atom to find $$Ca (40.078) + C (12.011) + O_3(15.99 \cdot 3) = 100.086 g$$. Now we can divide the mass of the cube by the molar mass of $$CaCO_3$$ to find the number of moles present. With this information, we can multiply by Avogadro's number to find the number of molecules of $$CaCO_3$$. We find $$357.943 \cdot \frac{1\,mol\,CaCO_3}{100.086 g\,CaCO_3} \cdot \frac{6.022 \cdot 10^{23}\,molecules}{1\,mol}=2.15\cdot10^{24}\,molecules$$. Finally, we know that there are three atoms of oxygen for every mole of $$CaCO_3$$. So we must multiply the number of molecules by the number of oxygen atoms per molecule, and we find our final answer to be $$2.15 \cdot 10^{24}\,molecules\,CaCO_3 \cdot \frac{3\,oxygen\,atoms}{1\,molecule\,CaCO_3}=6.46\cdot10^{24}\,atoms\,of\,oxygen$$.

### Subject: Biology

How does lysozyme protect your eyes from infection? Precisely what does it do? Which bacteria is more likely to cause an eye infection, Gram positive or Gram negative and why?

Lysozyme is found in tears; it protects your eyes from infection by breaking the $$\beta$$ $$1, 4$$ bonds that lend structure to the peptidoglycan in bacteria. Gram positive bacteria rely heavily on a thick wall of peptidoglycan, but Gram negative bacteria also have a protective membrane. This makes Gram negative bacteria better at causing eye infections.

### Subject: Calculus

Find the equation of the tangent plane to $$z=2e^{x}+3sin(y)$$ at the point $$(0, 0, 2)$$

We know that $$z-z_{0}=f_{x}(x_{0}, y_{0}) + f_0 (x_0, y_0)(y-y_0)$$. We derive to find that $$f_x=2e^x$$ and $$f_y=2y$$. At the point $$(0, 0, 2)$$, $$f_{{x}_{0}}=2$$ and $$f_{{y}_{0}}=3$$. By plugging into the original equation, we find that $$z-2=z(x-0)+3(y-0)$$. With some simplification, this becomes $$z=2x+3y+2$$.

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