A is equal to b squared plus four. Find a when b is five. ( a= b2 + 4 )
2 answers:
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Answer:
- The solution that optimizes the profit is producing 0 small lifts and 50 large lifts.
- Below are all the steps explained in detail.
- The graph is attached.
Explanation:
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<u>1. Name the variables:</u>
- x: number of smaller lifts
- y: number of larger lifts
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<u>2. Build a table to determine the number of hours each lift requires from each department:</u>
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Number of hours
small lift large lift total per department
Welding department 1x 3y x + 3y
Packaging department 2x 1y 2x + y
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<u>3. Constraints</u>
- 150 hours available in welding: x + 3y ≤ 150
- 120 hours available in packaging: 2x + y ≤ 120
- The variables cannot be negative: x ≥ 0, and y ≥ 0
Then you must:
- draw the lines and regions defined by each constraint
- determine the region of solution that satisfies all the constraints
- determine the vertices of the solution region
- test the profit function for each of the vertices. The vertex that gives the greatest profit is the solution (the number of each tupe that should be produced to maximize profits)
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<u>4. Graph</u>
See the graph attached.
Here is how you draw it.
- x + 3y ≤ 150
- draw the line x + 3y = 150 (a solid line because it is included in the solution set)
- shade the region below and to the left of the line
- 2x + y ≤ 120
- draw the line 2x + y ≤ 120 (a solid line because it is included in the solution set)
- shade the region below and to the left of the line
- x ≥ 0 and y ≥ 0: means that only the first quadrant is considered
- the solution region is the intersection of the regions described above.
- take the points that are vertices inside the solutoin region.
<u>5. Test the profit function for each vertex</u>
The profit function is P(x,y) = 25x + 90y
The vertices shown in the graph are:
- (0,0)
- (0,50)
- (42,36)
- (60,0)
The profits with the vertices are:
- P(0,0) = 0
- P(0,50) = 25(0) + 90(50) = 4,500
- P(42,36) = 25(42) + 90(36) = 4,290
- P(60,0) = 25(60) + 90(0) = 1,500
Thus, the solution that optimizes the profit is producing 0 smaller lifts and 90 larger lifts.
Answer:
The maximum possible number of mosquitoes is 25 millions
Step-by-step explanation:
Let
m(x) -----> the number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes)
x ----> the rainfall in centimeters
we have
This is a quadratic equation (vertical parabola) open downward
The vertex is the maximum
we know that
The maximum possible number of mosquitoes is equal to the y-coordinate of the vertex
Find out the coordinate of the vertex
The general equation of a vertical parabola in vertex form is equal to
where
a is a coefficient
(h,k) is the vertex
so
In this problem
a=-1 (open downward)
(h,k)=(5,25)
The y-coordinate is 25
therefore
The maximum possible number of mosquitoes is 25 millions