Answer :
To find the amount of radioactive material left in the vault after 140 years, we can use the provided function:
[tex]\[ f(x) = 200(0.5)^{x/50} \][/tex]
Here, [tex]\( x \)[/tex] is the number of years since the material was put into the vault. We want to find [tex]\( f(140) \)[/tex].
1. Substitute [tex]\( x = 140 \)[/tex] into the function:
[tex]\[ f(140) = 200(0.5)^{140/50} \][/tex]
2. Calculate the exponent:
[tex]\[ 140/50 = 2.8 \][/tex]
3. Calculate the power of 0.5:
[tex]\[ (0.5)^{2.8} \][/tex]
4. Multiply by the initial amount:
[tex]\[ f(140) = 200 \times (0.5)^{2.8} \][/tex]
5. Round to the nearest whole number:
After evaluating the expression, the function gives us approximately 28.72 pounds. When we round 28.72 to the nearest whole number, we get 29 pounds.
Therefore, the amount of radioactive material in the vault after 140 years is approximately 29 pounds.
[tex]\[ f(x) = 200(0.5)^{x/50} \][/tex]
Here, [tex]\( x \)[/tex] is the number of years since the material was put into the vault. We want to find [tex]\( f(140) \)[/tex].
1. Substitute [tex]\( x = 140 \)[/tex] into the function:
[tex]\[ f(140) = 200(0.5)^{140/50} \][/tex]
2. Calculate the exponent:
[tex]\[ 140/50 = 2.8 \][/tex]
3. Calculate the power of 0.5:
[tex]\[ (0.5)^{2.8} \][/tex]
4. Multiply by the initial amount:
[tex]\[ f(140) = 200 \times (0.5)^{2.8} \][/tex]
5. Round to the nearest whole number:
After evaluating the expression, the function gives us approximately 28.72 pounds. When we round 28.72 to the nearest whole number, we get 29 pounds.
Therefore, the amount of radioactive material in the vault after 140 years is approximately 29 pounds.
