Write each set in set-builder notation.

$$M = \{ 1,3,5,7,9,11,13,17,19 \}$$

A vitreous silica optical fiber of diameter $100 \mu \mathrm{m}$ is used to send optical signals from a sensor placed deep inside a hydrogen chamber. The hydrogen is at a pressure of 20 bars. The mass diffusivity and solubility of the hydrogen in the glass fiber are $D_{\mathrm{AB}}=2.88 \times 10^{-15}\mathrm{m}^{2} / \mathrm{s}$ and $S=4.15 \times 10^{-3} \mathrm{kmol} / \mathrm{m}^{3} \cdot \mathrm { bar }$, respectively. Hydrogen diffusion into the fiber is undesirable, since it changes the spectral transmissivity and refractive index of the glass and can lead to failure of the detection system. (a) Determine the average hydrogen concentration in an uncoated optical fiber, $\bar{C}$, after 100 h of operation in the hydrogen environment. Determine the corresponding change in the refractive index, $\Delta n$, of the fiber. For vitreous silica, $\Delta n=\left(1.6 \times 10^{-3} \mathrm{m}^{3} / \mathrm{kmol}\right) \times \bar{C}$. (b) Determine the average hydrogen concentration and change in refractive index after 1 h and 10 h of operation in the hydrogen environment.

A man owns two building supply stores, one on the east side and one on the west side of a city. Two customers order some $\frac{1}{2}$-inch plywood. Customer A needs 50 sheets, and customer B needs 70 sheets. The east-side store has 80 sheets, and the west-side store has 45 sheets of this plywood in stock. The east-side store’s delivery costs per sheet are $0.50 to customer A and$0.60 to customer B. The west-side store’s delivery costs per sheet are $0.40 to customer A and$0.55 to customer B. How many sheets should be shipped from each store to each customer to minimize delivery costs?