Grade 10 Academic MPM 2 D Unit 4

Quadratic Formulas Optimization Problems (More) © 2017 E. Choi – MPM 2 D -

Summary: Quadratic Formulas Vertex Form: Vertex: Axis of Symmetry: x-intercepts: y-intercept: Standard Form: Vertex:

Example 1: Determine the Vertex from Standard Form i) Convert the following quadratic relations

Example 2: Optimization Problem (Numbers) Find two real numbers x and y that sum

Example 3: Optimization Problems Last year the yearbook at Central High cost $75 and

Example 4: Optimization Problems Mary wants to fence a rectangular garden to keep the

Example 5: Optimization Problems A piece of wire 20 feet long is cut into

Homework Work sheet: Optimization Problems (Finish the word problems) Text: Check the website for

End of lesson Optimization Problems (More) © 2017 E. Choi – MPM 2 D

Slides: 11

Download presentation

Summary: Quadratic Formulas Vertex Form: Vertex: Axis of Symmetry: x-intercepts: y-intercept: Standard Form: Vertex: Axis of Symmetry: x-intercepts: y-intercept: Optimization Problems (More) © 2017 E. Choi – MPM 2 D - All Rights Reserved

Example 1: Determine the Vertex from Standard Form i) Convert the following quadratic relations to vertex form to determine the ii) vertex. Use the short cut (vertex formula) to determine the vertex from standard form. Optimization Problems (More) © 2017 E. Choi – MPM 2 D - All Rights Reserved

Example 2: Optimization Problem (Numbers) Find two real numbers x and y that sum to 50 and that have a product that is a maximum. Let x and y be the two numbers Therefore maximum P be the product of the two numbers product is 625 when the 2 numbers are both 25. Maximize Product: Given: Roots: Method 2: By Completing the square: Objective Relation Factored form!! Opens down, max value occurs! Axis of symmetry: Vertex: Optimization Problems (More) +625 -625 Vertex: Maximum value is 625 when x is 25. © 2017 E. Choi – MPM 2 D - All Rights Reserved

Example 3: Optimization Problems Last year the yearbook at Central High cost $75 and only 500 were sold. A student survey found that for every $5 reduction in price, 100 more students will buy yearbooks. What price should be charged to maximize the revenue from yearbook sales? History: $75 Sold 500 Let x be number of $5 changed Now: $5 100 + Revenue Maximum revenue is $50000 when $5 being reduced 5 times from $75 to $75 -5($5) = $50 1000 yearbooks will be sold. Vertex: Optimization Problems (More) © 2017 E. Choi – MPM 2 D - All Rights Reserved

Example 4: Optimization Problems Mary wants to fence a rectangular garden to keep the deer from eating her fruit and vegetables. One side of her garden attaches to her shed wall so she will not need to fence that side. However, she also wants to use material to separate the rectangular garden in two sections. She can afford to buy 80 metres of fencing to use for the perimeter and the section dividing the rectangular garden. What dimensions will maximize the SHED total area of the rectangular garden? x x x Let x and y be the dimensions of the garden in metres Given: y Area: Vertex: Optimization Problems (More) © 2017 E. Choi – MPM 2 D - All Rights Reserved

Example 5: Optimization Problems A piece of wire 20 feet long is cut into two pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. x y Let x and y be the lengths of the two pieces of wire in Square 1 feets Square 2 Vertex: (10, 12. 5) Minimum total area is 12. 5 ft 2 when the lengths of both wires are 10 feets. Optimization Problems (More) © 2017 E. Choi – MPM 2 D - All Rights Reserved