The local salon offers floral hairbands for adults and for children. The salon manager can make a total of 50 hairbands each day. The adult hairbands use 5 flowers and the children hairbands only use 3 flowers. There are 75 flowers available. The adult hairbands give an income of $5.00 and the children hairbands give an income of $3.00. How many of each of the adult hairbands and children hairbands should be made to maximize income. Solution needs to include the following: Objective Function Constraints Graph Vertices
in Word Problem Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Plot two lines on the same graph: A=50-C (constraint on numbers of hair bands) and A=(75-3C)/5 (from 5A+3C=75, the flower constraint), where A=number of adult bands and C=the number of child bands. The area under the first graph (a straight line joining the A intercept 50 and C intercept 50) includes the area under the second graph (A intercept 15, C intercept 25), so the more severe constraint is the second graph. Its area is enclosed by the axes and the line. The income equation R (revenue)=5A+3C matches the flower constraint, as it happens.

No matter how many bands of either type are made the income will always be $75 if all the available flowers are used. So 15 adult bands will fetch 15*5=$75 as will 25 child bands=25*3=$75. If we take 9 adult bands, for example, we need 45 flowers, leaving 30 flowers from which 10 child bands can be made. The income is $45+$30=$75. The total number of bands is 19, well below the capacity of 50. The number of child bands will always be a multiple of 5 and the number of adult bands a multiple of 3 to maximise income at $75 (C=0, 5, 10, 15, 20, 25 with A=15, 12, 9, 6, 3, 0, so A+C=15, 17, 19, 21, 23, 25).

The graph vertices are defined by the line A=(75-3C)/5 as (C,A)=(0,15), (0,0) and (25,0). These vertices enclose an area which is enclosed by the vertices (0,50), (0,0) and (50,0). These cover the constraints outlined in the question.

by Top Rated User (1.1m points)
edited by

Related questions

1 answer
asked May 7, 2020 in Algebra 1 Answers by lastguru Level 1 User (120 points) | 1.5k views
1 answer
2 answers
asked Feb 2, 2016 in Algebra 1 Answers by Jen | 945 views
1 answer
2 answers
asked Oct 5, 2014 in Algebra 2 Answers by anonymous | 1.7k views
2 answers
1 answer
asked May 16, 2013 in Geometry Answers by anonymous | 464 views
1 answer
asked May 14, 2013 in Algebra 1 Answers by anonymous | 484 views
1 answer
asked Apr 18, 2013 in Algebra 1 Answers by anonymous | 818 views
1 answer
asked Mar 27, 2013 in Geometry Answers by anonymous | 586 views
1 answer
asked Mar 26, 2013 in Algebra 2 Answers by anonymous | 419 views
1 answer
asked Mar 6, 2013 in Algebra 2 Answers by anonymous | 1.2k views
1 answer
asked Mar 5, 2013 in Algebra 1 Answers by anonymous | 536 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users