Question

College algebra questions

Answer 1

1) He originally bought 30 pigs.

2) 30/7 hours

Explanation:

Question 1

Let
`x` be the number of pigs the farmer bought.

If the farmer paid $210 for
`x` pigs, then the cost per pig was:

`\textsf{Cost per pig}=(\$210)/(x)`

If he made $8 of profit per pig sold, then the selling price of each pig was:

`\textsf{Selling price per pig}=(\$210)/(x)+\$8`

If he paid $210 for the pigs and his total profit was $150, then the total income from selling the pigs was:

`\textsf{Total income}=\$210+\$150=\$360`

If 6 pigs died before he could sell the rest, the number of pigs sold was:

`\textsf{Number of pigs sold}=x-6`

So, the total income from selling the pigs is equal to the number of pigs sold multiplied by the selling price per pig:

`360=(x-6)\cdot \left((210)/(x)+8\right)`

To determine the number of pigs the farmer original bought, solve the equation for x:

`(x-6)\cdot \left((210+8x)/(x)\right)=360`

`((x-6)(210+8x))/(x)=360`

`(x-6)(210+8x)=360x`

`8x^2-198x-1260=0`

`2(4x^2-99x-630)=0`

`4x^2-99x-630=0`

`4x^2-120x+21x-630=0`

`4x(x-30)+21(x-30)=0`

`(4x+21)(x-30)=0`

`x=-(21)/(4),\quad x=30`

As the number of pigs originally sold cannot be negative, x = 30 is the only valid solution. Therefore, the farmer originally bought 30 pigs.

`\hrulefill`

Question 2

Let K be Kaleb's work rate (the amount of work he can do per hour).

Let T be Trashia's work rate (the amount of work she can do per hour).

Kaleb completes the job in 10 hours, so his work rate is:

`K = (1)/(10)\;\textsf{of the job per hour}`

Working together, Kaleb and Trashia can complete the job in 3 hours, so their combined work rate is:

`K + T = (1)/(3)\;\textsf{of the job per hour}`

To find Trashia's work rate (T), substitute K = 1/10 into the equation for K + T:

`(1)/(10) + T = (1)/(3)`

Now, solve for T:

`(1)/(10) + T-(1)/(10) = (1)/(3)-(1)/(10)`

`T = (1)/(3)-(1)/(10)`

`T = (1\cdot 10)/(3\cdot 10)-(1\cdot 3)/(10\cdot 3)`

`T = (10)/(30)-(3)/(30)`

`T = (10-3)/(30)`

`T = (7)/(30)`

So, Trashia's work rate is 7/30 of the job per hour.

Now, to find how long it would take Trashia to do the job alone, take the reciprocal of her work rate:

`\textsf{Time} = (1)/(T) = (1)/((7)/(30))=(30)/(7)\sf \; hours`

Therefore, Trashia would take 30/7 hours to complete the job on her own.