Direct and Inverse Variation

Transcription

1 Lesson 72 Direct and Inverse Variation Review: Lesson 6, 28, 60, 71 72A Direct Variation God designed His creation to reflect His attribute of unity and diversity. One way He reveals this attribute to us is in the relationship between one thing when compared to another. For example, one way that we identify metals is by their density (Lesson 60). If I had a metal, and I told you it was pure gold, how would you know whether it was pure gold, and not brass, or fool s gold, or just gold-plated? You could measure its mass and volume, but neither of these measurements by themselves would identify it. However, when you divide its mass by its volume and calculate its density, then you have a value that will help you identify the metal. If it s density was 19.3 g/ml at room temperature, then you would know it was pure gold. But if it s density was about 5.0 g/ml, then I probably gave you fool s gold! Why are we talking about the density of gold? Well, the density of gold, the ratio of it s mass to volume, is constant. In fact, most elements and compounds have fairly constant densities, which means a direct variation exists between their mass and volume. As their mass increases, their volume increases, and the ratio is constant. Likewise, as their mass decreases, their volume decreases. Saying there is a direct variation between two things is no different than saying they are proportional to each other, because the ratio of their values is constant. In a word problem, look for key phrases like varies directly and directly proportional to, which will help you set the problem up correctly. Example 72.1 The mass (M) of gold varies directly with its volume (V). At room temperature, 19.3 g has a volume of 1 ml. What is the mass of 15 ml of gold? Round answer to 1 d.p. solution: The key phrase varies directly should immediately make you think To solve this, I need to set up a proportion. The ratio of M to V is constant, so M 1 V 1 = M 2 V 2 You have values for all but M 2, so rearrange and evaluate, just like you did with the scientific formula problems in Lesson 60B: M 2 = M V 1 2 = 19.3(15) = g V 1 1

2 In Lesson 60B, you were given a formula, but in this lesson, you have to create the formula based on key phrases given to you. So, pay attention to those key phrases! Example 72.2 At constant volume, the tank s air pressure (P) was directly proportional to its temperature (T). Initially, the pressure was 15 psi at a temperature of 300 Kelvins. What would the temperature be at a pressure of 20 psi? Round to the nearest whole number. solution: The key phrase directly proportional to should immediately make you think To solve this, I need to set up a proportion. The ratio of P to T is constant, so T 1 = P 2 Since the ratio is constant, you could also write T 1 = P 2. You have values for all but, so rearrange and evaluate: = T 2 = 300(20) = 400 K 15 Don t forget to write your units as well as the number! 72B Inverse Variation With direct variation, as one thing increases, so does the other, and as one thing decreases, so does the other. The ratio is a constant. Knowing that, you probably already know what inverse variation is about! With inverse variation, as one thing increases, the other decreases. Their product is constant. In word problems, when you see phrases like varies inversely or inversely proportional to, you will know to create a formula where the products are equal. You have already done some problems where two values were inversely proportional to each other, including Example 60.4 and Practice Set Example 72.3 In a first-class lever like a see-saw, the force (F) varies inversely with distance (d) from the pivot point, or fulcrum. On one side of the see-saw, a force of 10 N was applied at a distance of 3 m from the fulcrum. In order to balance the see-saw, how far from the fulcrum must a 6 N force be placed on the other side? solution: If you don t know what a see-saw is, refer to the example above.

3 The key phrase varies inversely should immediately make you think To solve this, I need to set up equal products of force times distance. The product of F and d is constant, so Rearrange and evaluate for d 2 : F 1 d 1 = F 2 d 2 d 2 = Fd 1 1 == 10(3) = 30 F = 5 m Take a minute to review Example Ex was the same problem, just worded differently! In mathematics, there are different ways to describe similar things. It s important for you to realize this, because it will help you avoid confusion on problems that may look difficult, but are actually quite easy. Example 72.4 Rate (R) is inversely proportional to time (T). Driving for 4 hours at a rate of 60 miles per hour is the same as driving 5 hours at what rate? solution: The key phrase inversely proportional to should immediately make you think To solve this, I need to set up equal products of rate times time. Rearrange and evaluate for R 2 : R 2 = RT 1 1 = 4(60) = 48 mph 5 R 1 T 1 = R 2 The most important thing to remember from this lesson is this: If you have a direct variation, set up equal ratios, and when you have an inverse variation, set up equal products. Practice Set 72 Use your best judgement as to when you should and shouldn t use a calculator (Natural history) Thick layers of gypsum (calcium sulfate) exist both above and beneath the ground in many places. Some believe the layers formed through evaportation of seawater, but the massive amounts of seawater required seem excessive for this to be a valid hypothesis. For example, the amount of seawater (S) required varies directly with the amount of gypsum (G) deposited. Depositing 1 foot of gypsum requires the evaporation of 1,400 feet of seawater. How many feet of seawater would be required to evaporate in order to create a 40-foot thick layer of gypsum? The air speed (S) in the wind tunnel was inversely proportional to the tunnel s cross-sectional area (A). At one point in the wind tunnel with an area of 0.2 m 2, the wind speed was 100 m/s. What would the wind speed be when the area increased to 1.0 m 2?

4 3 71. Solve for x. Begin by isolating the radical expression. x = Solve for x. Write answer as an improper fraction. x 2 x + 1 5x = Use elimination to solve the following system of equations. 4x y = 13 Write answer as an ordered pair. 3x + y = (Geology) To pay for an x-ray diffraction (XRD) system, the exploration company borrowed $50,000. Calculate their monthly payments, if their interest rate was 4%, compounded discretely, for 5 years Given AC BC C AD BD Prove ACD BCD A D B Given the circle and intersecting chords shown, write a two-column proof to show x = 4. Use the intersecting chords theorem in your proof x Evaluate. 1 x lim x 1 x Use substitution or elimination to solve. Write answer as an ordered pair. y = x 2 y = 4x Convert the point (1,3) to polar form. Round results to 2 decimal places (Geology) The following is a graph of oceanic crust mass in grams, versus volume in ml. Estimate the difference in mass between 0 and 2 ml of crust. mass volume

5 Solve for a. 6a + 3b - 2 = Use 4 unit multipliers to convert 100 cm 2 to ft 2. Write answer rounded to 2 d.p (CLEP College Algebra) Which of the following equals x yz for all values of x, y, and z? A) xy z B) x (y+z) C) x y x z D) (x y ) z Use a calculator to subtract. Write the answer in scientific notation, rounded to 1 d.p Convert the following to scientific notation, rounded to 2 d.p Find f (x) when f(x) = 20x Find x. 48 x True or False. All whole numbers are also counting numbers.