The vector gives the number of units of two models of laptops produced by a company.The vector gives the prices (in dollars) of the two models of laptops, respectively.Use dot products to determine the revenue for these two laptops if the price of each is increased by 3.5%.
A)$8,186,667
B)$8,175,000
C)$8,280,000
D)$8,227,500
E)$8,105,000

Question

The vector  gives the number of units of two models of laptops produced by a company.The vector  gives the prices (in dollars) of the two models of laptops, respectively.Use dot products to determine the revenue for these two laptops if the price of each is increased by 3.5%.
A)$8,186,667 
B)$8,175,000 
C)$8,280,000 
D)$8,227,500 
E)$8,105,000

Quizplus · Accepted Answer

Let the original prices of the two models be $p_1$ and $p_2$. After increasing the prices by 3.5%, the new prices will be $1.035p_1$ and $1.035p_2$.

The revenue from selling the laptops can be calculated by taking the dot product of the two vectors:

$(11eacce6\_1f17\_5b4f\_9f49\_b1248adceac5\_TB7896\_11)\cdot(11eacce6\_1f17\_5b50\_9f49\_61297b504b8e\_TB7896\_11) = (x_1,y_1)\cdot(x_2,y_2) = x_1x_2 + y_1y_2$

Plugging in the values we get:

$(11eacce6\_1f17\_5b4f\_9f49\_b1248adceac5\_TB7896\_11)\cdot(11eacce6\_1f17\_5b50\_9f49\_61297b504b8e\_TB7896\_11) = (100000, 50000)\cdot(950, 1350) = (100000)(950) + (50000)(1350) = 98500000$

After increasing the prices by 3.5%, the new revenue will be:

$(11eacce6\_1f17\_5b4f\_9f49\_b1248adceac5\_TB7896\_11)\cdot(11eacce6\_1f17\_5b50\_9f49\_61297b504b8e\_TB7896\_11) = (x_1,y_1)\cdot(1.035x_2,1.035y_2) = x_1(1.035)x_2 + y_1(1.035)y_2 = 1.035(x_1x_2 + y_1y_2)$

Plugging in the values we get:

$(11eacce6\_1f17\_5b4f\_9f49\_b1248adceac5\_TB7896\_11)\cdot(11eacce6\_1f17\_5b50\_9f49\_61297b504b8e\_TB7896\_11) = (100000, 50000)\cdot(981.25, 1397.25) = (100000)(981.25) + (50000)(1397.25) = 102300000$

Therefore, the revenue has increased from $98,500,000 to $102,300,000, an increase of $3,800,000. Thus, the revenue for the two laptops if the price of each is increased by 3.5% is $102,300,000.

The best choice is (C).

The Vector Gives the Number of Units of Two