Karen is buying pens and pencils for the new school year. She wants to have no more than 25 writing utensils in all. She also wa
nts the number of pencils to be greater than or equal to the square of 3 less than the number of pens. Create a system of inequalities to model the situation above, and use it to determine how many of the solutions are viable.
1 answer:
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Answer:
Step-by-step explanation:
To solve this problem, we will use the following two theorems/definitions:
- Given a vector field F of the form (P(x,y,z),Q(x,y,z),W(x,y,z)) then the divergence of F denoted by
- (Gauss' theorem)Given a closed surface S, the following applies
where n is the normal vector pointing outward of the surface and V is the volume bounded by the surface S.
Let us, in our case, calculate the divergence of the given field. We have that
Hence, by the Gauss theorem we have that
So, we must calculate the volume V bounded by the cube S.
We know that the vertices are located on the given points. We must determine the lenght of the side of the cube. To do so, we will take two vertices that are on the some side and whose coordinates differ in only one coordinate. Then, we will calculate the distance between the vertices and that is the lenght of the side.
Take the vertices (1,1,1) and (1,1-1). The distance between them is given by
.
Hence, the volume of V is . Then, the final answer is