please explain how to get these answers (answers provided):

1. If this cell were a muscle cell (myocyte), how many molecules of actin could it hold? (Assume the cell is spherical and no other cellular components are present; actin molecules are spherical, with a diameter of 3.6 nm. The volume of a sphere is 4/3 πr3

Answer: 2.7 X 10^12 actin molecules

2. If this were a liver cell (hepatocyte) of the same dimensions, how many mitochondria could it hold? (Assume the cell is spherical; no othe cellular components are present; and the mitochondria are spherical, with a diameter of 1.5 μm).

Answer: 36000 motochondria

3. Glucose is a major energy-yielding nutrient for most cells. Assuming a cellular concentration of 0f 1 mM, calculate how many molecules of glucose would be present in our hypothetical (and spherical) eukaryotic cell. (Avogadro's number, the number of molecules in 1 mol of a nonionized substance, is 6.02 X 10^23).

Answer: 3.9 X 10^10 glucose molecules

4. Hexokinase is an important enzyme in the metabolism of glucose. If the concentration of hexokinase in our eukaryotic cell is 20 μM, how many glucose molecules are present per hexokinase molecule?

Answer:50 glucose molecules per hexokinase molecule

5. E. coli cells are rod-shaped, about 2 μm long and 0.8 μm in diameter. The volume of a cylinder is πr2h, where h is the height of the cylinder.

A) if the average density of E. coli (mostly water) is 1.1 X 103 g/L, what is the mass of a single cell?
Answer: 1 X 10^-12 g = 1 pg

B) E. coli has a protective cell envelope 10 nm thick. What percentage of the total of the bacterium does the cell envelope occupy?
Answer: 10%

C) E. coli is capable of growing and multiplying rapidly because it contains some 15000 spherical ribosomes (diameter 18 nm), which carry out protein synthesis. What percentage of the cell volume do the ribosomes occupy?
Answer: 5%

1) There's no way to do this without knowing the volume of the myocyte, so I would recommend you to search the book or your notes, should be somewhere in there. Basically, you would approach this question by finding the volume of the actin molecule first. V = 4/3*pi*(1.8nm)^3 = 24.42nm^3. Then your next step is just take the volume of the myocyte divide by 24.42nm^3 and you will get the number of molecules of actin inside a single myocyte (make sure the volume for myocyte is in nm^3, if not, you will need to do some unit conversions so the units can cancel out).

And just a hint or you can call it a cheat, lol. Since you already gotten the answer, if you don't want to go through the hassle of digging through your book and notes for the volume of the myocyte, you can just go 2.7x10^12 = x/24.42nm^3. Solve for x and that's the volume of the myocyte in nm^3. However, you wouldn't be able to do this if you do not know the answer ahead of time, so it's up to you. It doesn't really matter because in an exam, I'm sure your teacher would provide that value for you and won't make you memorize it.

Yeah for the rest of these, you just need to know the volume of the cells to do the math.

6th textbook

2. Components of E. coli E. coli cells are rod-shaped, about 2 _m long and 0.8 _m in diameter. The volume of a cylinder is _r2h, where h is the height of the cylinder.
(a) If the average density of E. coli (mostly water) is 1.1 3 103 g/L, what is the mass of a single cell?
Ans:It’s diameter is 0.8 μm so radium is 0.4 μm. Volume is pir^2h.
V=πr^2 h=π×〖0.4〗^2×2=0.32π 〖μm〗^3
The number of E. coli bacteria is 1L/V.
N=(1 L)/(0.32π 〖μm〗^3 )=(1 〖dm〗^3)/(0.32π 〖μm〗^3 )=(〖10〗^15 〖dm〗^3)/(0.32π 〖μm〗^3 )=9.94718 × 〖10〗^14
The mass of single cell is M⁄N=(1.1×〖10〗^3 g)⁄(9.94718 × 〖10〗^14 )≈〖10〗^(-12) g.
(b) E. coli has a protective cell envelope 10 nm thick. What percentage of the total volume of the bacterium does the cell envelope occupy?
Ans: The volume of a E. coli bacterium except cell envelope should use smaller radium as 0.39μm.
V'=π〖r^'〗^2 h=π×〖0.39〗^2×1.98=0.301158π 〖μm〗^3
Cell envelope occupies 1-0.301158π⁄0.32π=5.89%.
(c) E. coli is capable of growing and multiplying rapidly because it contains some 15,000 spherical ribosomes (diameter 18 nm), which carry out protein synthesis. What percentage of the cell volume do the ribosomes occupy?
Ans: We regard ribosome as ball. One ribosome volume is V=4/3 πr^3=4/3 π〖(9nm)〗^3=972π 〖nm〗^3.
The ribosomes volume is 15,000V=15,000×972π 〖nm〗^3=0.0458 〖μm〗^3 . They occupy 15,000V⁄0.32π=4.55625%

To get the answers to these questions, you need to follow the given information and use the appropriate formulas or calculations. Here's how you can arrive at each answer:

1. To find out how many molecules of actin a muscle cell can hold, you can use the formula for the volume of a sphere (V = 4/3 πr^3) and the diameter of the actin molecule (d = 3.6 nm). You can assume the cell is spherical and use the given cell dimensions to calculate the radius (r) of the cell. Then, divide the volume of the cell by the volume of a single actin molecule (V_actin = 4/3 π(1.8^3)). The result will give you the number of actin molecules a muscle cell can hold.

2. Similar to the previous question, you can use the formula for the volume of a sphere to calculate the volume of the liver cell using its dimensions. Then, divide the volume of the cell by the volume of a single mitochondrion (V_mitochondrion = 4/3 π(0.75^3)). The result will give you the number of mitochondria a liver cell can hold.

3. For this question, you need to calculate the volume of the hypothetical eukaryotic cell using the given formula for the volume of a sphere. Then, using the cellular concentration of glucose (1 mM), you can convert it to moles of glucose using Avogadro's number. Finally, multiply the number of moles of glucose by Avogadro's number to get the number of glucose molecules in the cell.

4. To find out how many glucose molecules are present per hexokinase molecule, you need to divide the concentration of hexokinase (20 μM) by the concentration of glucose molecules (1 mM), keeping in mind the appropriate conversion factors to match the units. The resulting ratio will give you the number of glucose molecules per hexokinase molecule.

5. For this set of questions, you need to use the formulas for the volume of a cylinder and the volume of a sphere to calculate the mass and volume of the E. coli cell and its components. For part A, you multiply the volume of the E. coli cell (calculated from the given dimensions) by the density of E. coli to obtain the mass of a single cell. In part B, you calculate the additional volume occupied by the cell envelope (10 nm thick), divide it by the total volume of the bacterium, and multiply by 100 to get the percentage. In part C, you take the number of ribosomes (15000) and calculate the total volume they occupy by considering them as spheres. Then, you divide this volume by the total volume of the cell and multiply by 100 to get the percentage.