Section 4 7 Forming Functions from Verbal Descriptions

Section 4 -7 Forming Functions from Verbal Descriptions Objectives: v To form a function of one variable from a verbal description § To determine the minimum or maximum value of that function

Introduction • Finding Minimum and Maximum values of a function is an IMPORTANT part of mathematics. • Ex) When engineers and architects design a building, they must: • Minimize cost • Maximize production • Minimize structural stress on a girder • Maximize volume from a given amount of material • Recall: we can find Min’s and Max’s using a graphing utility or using calculus (vertex of a parabola) – Note: Min’s and Max’s are known as extreme values • KEY – Before we can find a max or a min, we need to create a function (or a rule) that models (fits) the situation – Often we will create a function of 2 variables , but in order to solve we must reduce the function to ONE variable.

What we are given • Example 1: An open-top box with a square base is to be constructed from sheet metal in such a way that the completed box is made of 2 m 2 of sheet metal. Express the volume of the box as a function of the base width. General Problem Solving Strategy What to Find What we are given Create a visual – Diagrams are often very helpful We know the box is open (no top) We know the box has a square base (all base lengths are the same) We do not know the height h w w Now let’s try to express the volume of the box as a function of the base width Since we have a square base We know Volume = L*W*H Volume = W*W*H = W 2*H

Example 1 Continued… Recall: KEY: Always try to obtain a function of ONE variable Volume = W*W*H = W 2*H Since Volume is a function of width and height, we write Problem: we have volume as a function of two variables, we want volume to be a function of ONLY width. Recall: “Express the volume of the box as a function of the base width. ” Let’s try to use more of the given information: 2 m 2 of sheet metal (this is a measure of Area) h open-top box Area of sheeting used = 2 Area of Base + Area of Sides = 2 + (No top) =2 (4 sides) Now we solve for h (to get a function of w) w w To find V in terms of w alone, we substitute

Example 1 Continued… If we were asked to maximize the volume of the open box, given 2 m 2 of material we could do it. Moving from a function of two variables (width and height) to a function of one variable (width) is a HUGE STEP. This allows us to graph the function and find a maximum Recall: V w We maximize volume when w = ~0. 8 We know that width cannot be negative, so only look at the positive part (w>0) of the graph.

Example 2: A north-south bridle path intersects an east-west river at point O. At noon, a horse and rider leave O traveling north at 12 km/h. At the same time, a boat is 25 km east of O traveling west at 16 km/h. Express the distance d between the horse and the boat as a function of the time t in hours after noon. General Problem Solving Strategy What to Find path What we are given H = 12 t Create a visual – Diagrams are often very helpful A right triangle has developed in our diagram. d 16 t O 25 km B = 25 – 16 t Since the boat is 25 km from O and traveling toward O at 16 km/h Our goal is to find d (the hypotenuse), but we only know one side H. Approach: we can find the length of the other side and use the Pythagorean Theorem to find d.

Example 2 Continued… Follow up question: How close will the boater and the horse rider get to one another (i. e. what is the minimum distance between the two)? Recall: When this is a minimum, this will be a minimum We know this is a parabola, that catches water (400 is positive) t=1 The minimum always occurs at the vertex (where the graph intersects the axis of symmetry) axis of symmetry: (1, 15) So, the minimum distance between the horse and the boat is 15 km (and it happens at 1: 00 pm – 1 hr after their initial positions. )

Homework • P 161 -163: 1 -7 (odd), 11, 13, 14, 17, 19

Problem Solving Strategy UNDERSTAND the problem What am I trying to find? What data am I given? Have I ever solved a similar problem? Develop and carry out a PLAN What strategies might I use to solve the problem? How can I correctly carry out the strategies I selected? Find the ANSWER and CHECK Does the proposed solution check? What is the answer to the problem? Does the answer seem reasonable? Have I stated the answer clearly?