$1.$ Strontium$-90$ has a half$-$life of $28$ days. Suppose a sample has an initial mass of$25$ mg$.$

a$)$Find an expression for the mass of the sample of Strontium$-90$ after tt$$ days.

b$)$ Find the mass of the sample after$50$ days. Round to three decimal places.

c$)$ How long does it take for the sample to contain only$3$mg of Strontium$-90?$ Round to three decimal places.

$2.$ A bacteria culture grows with a constant relative growth rate. The bacteria count was $400$ after $2$ hours and $15,625$after $6$ hours.

a$)$ What is the relative growth rate? Enter the exact value for$$k$.$

b$)$ What was the$$initial size of the culture?$$

c$)$ Find an expression for the number of bacteria after t$$ hours.$$

d$)$ Find the number of cells after$33$ hours.

e$)$ When will the population reach $1,000,000?$ Round to three decimal places.

$3.$ The United States population in $1800$ was $5,308,000$ and in $19001900$ was $76,212,000.$ Assume t$=0$ in$1800$ and the population grew at a rate proportional to its size.

a$)$ Find P$($t$),$ the number of people in the United States at time tt according to this data. $($Use an exact value for the rate,$$k$.)$

b$)$ Using this model, predict the U$.$S$.$ population in the year $2000.$ Round to the nearest person.

$4.$ Suppose the position, in meters, of a particle moving along a horizontal line after tt$$ seconds can be modeled by the function

s$($t$)=5/4$t$^420/3$t$^330$t$^2+30$t t$0$

a$)$ Find the velocity, v$($t$)$ of the particle at time t$.$

b$)$ At what time$($s$)$ is the velocity of the particle $30$ m$/$s$?$ Separate multiple values with a comma.

c$)$ At what time$($s$)$ is the acceleration of the particle $0$ m$/$s$2?$ Separate multiple values with a comma.

$5.$ When a drink is taken from a refrigerator, its temperature is$4$C$.$ After$20$ minutes in a$25$C$$ room, its temperature has increased to$14$C

a$)$What is the temperature of the drink after$60$ minutes? Round to three decimal places.

b$)$ When will its temperature be$22$C$?$ Round to three decimal places.