The vector u = <3140, 2750> gives the number of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month.  The vector v = <2.25, 1.75> gives the prices (in...

The vector u = <3140, 2750> gives the number of hamburgers and hot dogs, respectively, sold at a fast-food stand in one month.  The vector v = <2.25, 1.75> gives the prices (in dollars) of the food items.

a)  Find the dot product u * v and interpret the result in the context of the problem.

b) Identify the vector operation used to increase the prices by 2.5%.

Expert Answers
Seema Adiga eNotes educator| Certified Educator

Lets rearrange the forms of the vectors

The number of hamburgers and hotdogs, 

The prices of them, `v = 2.25 i + 1.75 j `

a) The dot product of `u*v`  

 

`u*v = 3140X2.25 + 2750X1.75 `

     

`= 7065 + 4812.5 = $11877.5 `

b) 2.5% increase  in their price can be represented by `$1.025`

`[(100+2.5)/100 = 1.025]`

Second vector v has to be multiplied by the scalar k, i. e., 1.025

 

`therefore v = 1.025(2.25 i + 1.75 j) `

                 

`= 1.025X2.25i + 1.025X1.75j `

               

` = 2.30625 i + 1.79375j `

`= 2.31i + 1.79j `               (rounded up to two decimals)

The required vector is `kv(2.31, 1.79)`

 

 

Brandon Cohen | Student

To begin this problem, let's isolate the vectors we have been given and identify exactly what they represent.

Our first vector is vector u, which has a value of <3140,2750>. This vector identifies the number of hamburgers and hotdogs, respectively, that were sold.

The second vector is vector v, which has a value of <2.25,1.75>. This vector identifies the prices (in dollars) of the hamburgers and hotdogs, respectively.

By identifying exactly what information each vector provides us with, we can have a better understanding of the foundation, or building blocks, to this problem. 

a) In order to complete this problem, you must know what the dot product is. A simple representation of this is:

When `u=<u_(1),u_(2)> and v=<v_(1),v_(2)>`

then ` u*v=u_(1)v_(1)+u_(2)v_(2) `

With vector u, 3140 represents `u_(1)` , 2750 represents `u_(2)` . With vector v, 2.25 represents `v_(1)` , 1.75 represents `v_(2)` .

Now that we have identified how to find our dot product, we just calculate!

`u*v=(3140)(2.25)+(2750)(1.75)` 

` =7065+4,812.50 `

`=11,877.50`

In the context of this problem, when we solve for the dot product we are simply finding the total collected from food sales of hamburgers and hot dogs. 

b) On this part you are simply multiplying each part of vector v by 1.025. We use the number 1.025 to represent that there is an increase, which is 2.5% in this case. (Scalar Multiplication)

`v=<(2.25)(1.025),(1.75)(1.025)>`

`=<2.30625,1.79375>` This is an unsimplified form. If you want to have the correct number of significant figures, then round to 2 decimal places (based on the original vector).

`=<2.31,1.79>` This represents a 2.5% increase in the price of both hamburgers and hot dogs.

Another way you may see this written is:

`k*v, k=1.025`  In this case, k represents our scalar (2.5% increase)

`k*v=k<2.25,1.75>`

`=1.025<2.25,1.75>`

`=<(2.25)(1.025),(1.75)(1.025)>`

`=<2.30625,1.79375>`

` k*v=<2.31,1.79>`

` `

caressezhu | Student

For part a) of the problem, we know that the formula to calculate vector multiplication is u*v=u1*v1+u2*v2

So we have 3120*2.25+2750*1.75=11877.5

For part b), the price is increased by 2.5%. So we have <2.25*1.025, 1.75*1.025>, which is equal to <2.30625, 1.79375>.

We can write it as <2.3, 1.8>