1)
[tex]1.2 \; \text{g}[/tex]
There are [tex]75000/24000 = 25/8 = 3.125[/tex] half lives in a period of 75,000 years. The sample of plutonium-239 would thus expect a loss in mass that would results in a mass ratio of [tex](1/2)^{25/8}[/tex] which is the same as [tex]2^{-25/8} \approx 0.12 [/tex] relative to the initial mass. [tex]0.12 \times 10 \; \text{g} = 1.2 \; \text{g}[/tex] of plutonium-239 would thus remain in the initial [tex]10 \; \text{g}[/tex] sample after 75,000 years.
2)
[tex]1.5 \times 10^{3} \; \text{g}[/tex]
There are [tex]160/30 = 16/3 \approx 5.3[/tex] half lives in 160 years.
[tex]\text{mass remaining in 160 years} / \text{initial mass} = (1/2)^{160/30} \approx 2.5 \times 10^{-2}[/tex]
Thus
[tex]\text{initial mass} = 37 \; \text{g} / (2.5 \times 10^{-2})= {\text{initial mass} } \approx 1.5 \times 10^{3} \; \text{g}[/tex]
