Question

Sophia expects the number of cows, C C, on her farm t t years from now to be modeled by the function C ( t ) = 3 0 ( 2 ) C(t)=30(2) t . Additionally, she expects the supply of hay, F F, in tons, that her crops can provide for each cow t t years from now to be modeled by the function F ( t ) = 8 ( 1 . 5 ) F(t)=8(1.5) t . Let H H be the total yearly amount of hay produced in Sophia's farm (in tons) t t years from now. Note that the hay produced on Sophia's farm is used exclusively to feed her cows. Write the formula of H ( t ) H(t) in terms of C ( t ) C(t) and F ( t ) F(t).

Answer 1

The expression is given as:

`H(t)=C(t)* F(t)=240* 3^t`.

Explanation:

Sophia expects the number of cows, C, on her farm t years from now to be modeled by the function:

`C( t ) = 30* (2)^t`

Additionally, she expects the supply of hay, F, in tons, that her crops can provide for each cow t years from now to be modeled by the function

`F( t ) =8* (1.5)^t`

Let H be the total yearly amount of hay produced in Sophia's farm (in tons) t years from now.

Total amount of Hay produced in sophia's farm= Number of cows in farm×Amount of hay required for each cow.

i.e. H(t)=C(t)×F(t)

`H(t)=30* 8* (2)^t* (1.5)^t`

and we know that
`a^x* b^x=(a* b)^x=(ab)^x`

Hence,
`H(t)=240* (2* 1.5)^t\\\\H(t)=240* 3^t`.

Hence, the hay produced on Sophia's farm is used exclusively to feed her cows i.e. we need to write the formula of H ( t ) in terms of C(t) and F (t) is:

`H(t)=C(t)* F(t)=240* 3^t`.