Questions: UCSD is one of the first schools to create a department of NanoEngineering. The prefix "nano" refers to very small things and is based on a size definition from chemistry. A nanometer is 1 · 10^-7 meters. Similarly, a nanoliter is 1 · 10^-9 liters. Solve the below problems and write your answer in scientific notation. (a) The width of a human hair is about 80000 nanometers. Find the length in meters. (b) A triangular nanoparticle (often used as a catalyst or for drug delivery) has a base of 4 nanometers and height of 6 nanometers. Find the area in square meters. (c) A spherical virus has a radius of 20 nanometers. Find the approximate volume in cubic meters. Note that the volume, V, of a sphere with radius, r, is V = 4/3 π r^3. Use r ≈ 3 when doing your calculation.

Solution Steps

(a) To convert the width of a human hair from nanometers to meters, multiply the given width by the conversion factor \(1 \cdot 10^{-9}\).

(b) To find the area of a triangular nanoparticle in square meters, use the formula for the area of a triangle \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\) and then convert the result from square nanometers to square meters.

(c) To find the volume of a spherical virus in cubic meters, use the formula for the volume of a sphere \(V = \frac{4}{3} \pi r^3\) and then convert the result from cubic nanometers to cubic meters.

The width of a human hair is given as \( 80000 \) nanometers. To convert this to meters, we use the conversion factor \( 1 \, \text{nanometer} = 1 \cdot 10^{-9} \, \text{meters} \):

\[ \text{Width in meters} = 80000 \times 1 \cdot 10^{-9} = 8.00 \times 10^{-5} \, \text{meters} \]

The triangular nanoparticle has a base of \( 4 \) nanometers and a height of \( 6 \) nanometers. The area \( A \) of a triangle is calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Substituting the values:

\[ A = \frac{1}{2} \times 4 \times 6 = 12 \, \text{square nanometers} \]

To convert this to square meters, we use the conversion \( 1 \, \text{square nanometer} = 1 \cdot 10^{-18} \, \text{square meters} \):

\[ \text{Area in square meters} = 12 \times 1 \cdot 10^{-18} = 1.20 \times 10^{-17} \, \text{square meters} \]

The radius of the spherical virus is \( 20 \) nanometers. The volume \( V \) of a sphere is given by the formula:

\[ V = \frac{4}{3} \pi r^3 \]

Using \( \pi \approx 3 \) for the calculation:

\[ V = \frac{4}{3} \times 3 \times (20)^3 = \frac{4}{3} \times 3 \times 8000 = 33493.33 \, \text{cubic nanometers} \]

To convert this to cubic meters, we use the conversion \( 1 \, \text{cubic nanometer} = 1 \cdot 10^{-27} \, \text{cubic meters} \):

\[ \text{Volume in cubic meters} = 33493.33 \times 1 \cdot 10^{-27} = 3.35 \times 10^{-23} \, \text{cubic meters} \]

Final Answer