Lesson 4.1 (Part 1): Roots & Pythagorean Theorem

Transcription

1 Lesson 4.1 (Part 1): Roots & Pythagorean Theorem Objectives Students will understand how roots are used to undo powers. how the Pythagorean theorem is used in applications. Students will be able to use roots to solve power equations. solve problems using the Pythagorean theorem. Previously, we have explored another type of equation, a quadratic. This is a type of power equation. A power equation has the format y = x n. We have techniques to solve problems involving linear and exponential equations. Now we want to develop a technique for solving power equations. We need to find a way to undo power equations, ones of the form x n = b, where x is the input variable and n and b are both numbers, like x 2 = 9, and x 3 = 8. For this, we need roots. You ve probably seen the square root,, before. The square root is defined as the undoing of squaring. For example, since 3 2 = 9, then 9 = 3. Because of this, we can say that: x 2 = x, and ( x) 2 = x, so long as x is positive. For example: 4 2 = 16 = 4, and ( 25) 2 = 5 2 = 25. We can then use this to solve problems like x 2 = 81. But we do run into a small complication. Using the square root on both sides, we d get x 2 = 81, so x = 9. But notice that ( 9) 2 = 81 as well, so our technique of using the square root to undo squaring only got us one of the two answers. It is, unfortunately, common for an algebraic technique to only give one solution. We should check to see if there are other solutions. Solving equations like x 2 = b There are two solutions: x = b and x = b. Sometimes you ll see this written as x = ± b Example: Solve x 2 = 41 The solutions will be x = 41 and x = 41. There is no whole number whose square is 41, so we either have to leave the answer like this, or approximate the answer using a calculator, getting solutions x and x Sometimes the more precise exact answer is better. But in some real life problems, the approximation is more useful. In some cases it might be necessary to first isolate the x 2 by adding, subtracting, multiplying, or dividing.

2 Example: Solve x2 5 3 = 4 We need to isolate the x 2 first: Add 3 to both sides: x 2 5 = 7 Multiply both sides by 5: x 2 = 35 The solutions are x = 35 and x = Solve these equations: a. x 2 = 81 b. 3x 2 = 12 c. 5x = Recall that the formula for an area of a circle is A = πr 2. Use the technique of undoing the squaring to find the radius of a circle whose area is 300 square meters.

3 3. Let s look at the braking distance of a car. We can model the distance a car takes to brake using the equation d = V where V is the initial velocity of the car in feet per second, and d is the braking distance in feet. a. Suppose there is a car crash. Police can determine from skid marks that the car took 120 feet to brake. Determine the car s speed when it started braking. Give the speed in miles per hour. b. In a similar crash with the same road conditions, a car took 240 feet to brake. Do you think, based on your last answer, that the car s initial speed was: i. Less than double the first car s speed ii. Double the first car s speed iii. More than double the first car s speed c. Check if your intuition in part b was correct by calculating the speed.

4 Right triangles are triangles where one angle is a right angle (90 degree angle), like the corner of a square. Right triangles have a special property called the Pythagorean Theorem relating the length of the sides. If the two sides touching the right angle (called the legs of the triangle) are called a and b, and the long side (called the hypotenuse of the triangle) is called c, then they are related by the equation: a 2 + b 2 = c 2 This property is commonly used in construction and other professions. b c 4. If the legs of a triangle have lengths 6 ft and 8 ft, find the length of the hypotenuse. a 5. Someone wants to build a wheelchair ramp up to their front door. The ramp is 8 feet long. The side of the ramp against the house is 2 feet above the ground. How far out from the house will the end of the ramp be? Round your answer to two decimal places.

5 We can also solve other power equations, like x 3 = 8 and x 4 = 300, by using higher order roots. For example the cube root, 3, undoes cubing: = 8, so 8 = 2. In this case when we solve x 3 = 8, ( 2) 3 = 8, so there s only the one solution, x = 2. We end up getting two solutions when the power is even, and only one solution when the power is odd. When solving x n n = b when n is even, there are two solutions: x = b When solving x n n = b when n is odd, there is one solution: x = b n and x = b 6. Solve these equations: a. x 3 = 27 b. 4x 3 = 56 c. x 4 = An artist is creating a sculpture using a sphere made of clay to represent Earth. The volume of a sphere is given by the equation: V = 4 3 πr3, where r is the radius of the sphere. The artist has a rectangular slab of clay that is 4 inches wide, 6 inches long, and 2 inches high. What is the radius of the largest sphere the artist can create with this clay? 2 in 6 in 4 in