Question

George has opened a new store and he is monitoring its success closely. He has found that this store’s revenue each month can be modeled by r(x)=x2+5x+14 where x represents the number of months since the store opens the doors and r(x) is measured in hundreds of dollars. He has also found that his expenses each month can be modeled by c(x)=x2−3x+4 where x represents the number of months the store has been open and c(x) is measured in hundreds of dollars. What does (r−c)(3) mean about George's new store?

Answer 1

This is a great question!

When we are given ( r - c )( 3 ), we are being asked to take 3 as x in the functions r( x ) and c( x ), taking the difference of each afterwards -

$r( 3 ) = ( 3 )^2 + 5( 3 ) + 14,\\x( 3 ) = ( 3 )^2 - 3( 3 ) + 4$

____

Let us calculate the value of each function, determine their difference, and multiply by 100, considering r( x ) and c( x ) are measured in hundreds of dollars,

$r( 3 ) = 9 + 15 + 14 = 38,\\x( 3 ) = 9 - 9 + 4 = 0 + 4 = 4\\----------------\\( r - c )( 3 ) = 38 - 4 = 34,\\34 * 100 = 3,400( dollars )\\\\Solution = 3,400( dollars )$

Therefore, ( r - c )( 3 ) " means " that George's new store will have a profit of $3,400 after it's third month in business, given the following options,

( 1. The new store will have a profit of $3400 after its third month in business.  

( ​​2. The new store will have a profit of $2400 after its third month in business.

( 3. ​The new store will sell 2400 items in its third month in business.

( 4. The new store will sell 3400 items in its third month in business.

Answer 2

The required answer is , $(r-c)(5)$ means the revenue less expenses in 5 months i.e. the new store will have a profit of $ 5400 after 5 months.

Substitution:

The substitution method is the algebraic method to solve simultaneous linear equations.

Given function is,

$r(x) = x^2+5x+14$...(1)

And $c(x) = x^2-4x+5$...(2)

Now, substituting the value into the equation (1) and (2).

$r(5) = (5)^2+5(5)+14=64$

$c(5) = (5)^2-4(5)+5=10$

Therefore,

$(r-c)(5)=r(5)-c(5)\\=64-10\\=54$

Now, $(r-c)(5)$ means the revenue less expenses in 5 months i.e. the new store will have a profit of $ 5400 after 5 months.

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