2.6-Constructing Functions with Variation
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1 .6-Constructing Functions with Variation Direct Variation: Two quantities vary directly if one quantity increases as the other increases. A direct variation equation will have the general form: where is called the constant of proportionality. Eample: Suppose y varies directly with, and 6 when = 0. Find the constant of variation and the direct variation equation. Solution: We now this is a direct variation problem because we are told in the problem itself. The first step is to use the general direct variation equation to find by substituting the nown values of and y into this equation and simplifying. 6 = (0) 6 0 = = The constant of proportionality is =. Now we can find the direct variation equation by substituting the value of into the general form of the equation. y = Eample: Find the constant of variation and the direct variation equation in which y varies directly as and 0 when =. Solution: Again, we are told in the problem that the quantities vary directly so we will use the general direct variation equation. First find the value of, and then substitute this into the general equation. 0 = () =
2 Eample: Find the value of y when = if y varies directly with, and when = 6. Solution: First, find using the given information. = (6) = Now use the value of and the value of to solve for y. () 6 Therefore, 6 when =. Eample: Find the value of q when p = 9 if p varies directly as q, and p = when q =. Solution: First, find using the given information. Instead of and y, the equation will have p and q. p = q = () = Now use the value of and the value of p to solve for q. p = q 9 q = 9 p Therefore, q = when p = 9. 9 q = 9(9) q =
3 Inverse Variation: Two quantities vary inversely if one quantity increases as the other decreases. An inverse variation equation will have the general form: where is called the constant of proportionality. Notice that is still in the numerator but the other variable is now in the denominator. Eample: Suppose y varies inversely with, and when =. Find the constant of variation and the inverse variation equation. Solution: As before, we now this is an inverse variation problem because we are told in the problem itself. The first step is to use the general inverse variation equation to find by substituting the nown values of and y into this equation and simplifying. = = The constant of proportionality is =. Now we can find the inverse variation equation by substituting the value of into the general form of the equation. Eample: Find the constant of variation and the inverse variation equation in which y varies inversely with and when =. Solution: We are told in the problem that the quantities vary inversely so we will use the general inverse variation equation. First find the value of, and then substitute this into the general equation and simplify if necessary. = =
4 Eample: Find the value of y when = if y varies directly with, and when =. Solution: First, find using the given information. = = 0 Now use the value of and the value of to solve for y = 0 Therefore, 0 when =. Eample: Find the value of t when r = 9 if r varies inversely with t, and r = when t = 6. Solution: First, find using the given information. Instead of and y, the equation will have r and t. r = t Now use the value of and the value of r to solve for t. r = t t = r t = 9 Therefore, t = when r = 9. 9 = = 6
5 Joint Variation: Three or more quantities vary jointly if one quantity increases as the other quantities also increase. A joint variation equation will have the general form: where is called the constant of proportionality. Eample: Suppose y varies directly with and, and when = and = 6. Find the constant of variation and the joint variation equation. Solution: The first step is to use the general joint variation equation nown values of and y into this equation and simplifying. = ()(6) = = to find by substituting the The constant of proportionality is =. Now we can find the joint variation equation by substituting the value of into the general form of the equation. Eample: Find the constant of variation and the joint variation equation in which y varies directly as and, and, when = and =. Solution: The quantities vary jointly so we will use the general joint variation equation. First find the value of, and then substitute this into the general equation. = ()() =
6 Eample: Find the value of y when = and = if y varies directly with and, and when = 6 and =. Solution: First, find using the given information. = (6)() = Now use the value of and the values of and to solve for y. () 6 Therefore, 6 when = and =. Eample: Find the value of m when c = when f = and m =. c = and f 6 = if f varies jointly with c and m, and Solution: First, find using the given information. f = cm = () = Now use the value of and the value of c and f to solve for m. f = cm f m = c 6 m = 7 m = 7 Therefore, m =.
7 Combined Variation: Compound variation occurs when a quantity varies directly with one quantity and inversely with another. A compound variation equation will have the general form: where is called the constant of proportionality. Eample: Suppose y varies directly with and inversely with, and when = and =. Find the constant of variation and the combined variation equation. Solution: The first step is to use the general combined variation equation the nown values of and y into this equation and simplifying. () = = to find by substituting The constant of proportionality is =. Now we can find the combined variation equation by substituting the value of into the general form of the equation. Eample: Find the constant of variation and the combined variation equation in which y varies inversely with and directly with and when = and =. Solution: Use the general combined variation equation this into the general equation. () = =. First find the value of, and then substitute
8 Eample: Find the value of y when = and = if y varies directly with and inversely with, and y = when = and =. Solution: First, find using the given information. () = = Now use the value of and the values of and to solve for y. 6 0 Therefore, when = and =.
9 .6-Applications Eample: The pressure eerted by water at a point below the surface varies directly with the depth. The pressure is lb/sq in at a depth of ft. What pressure does the sperm whale eperience when it dives 6,000 ft below the surface? Solution: Since this is a direct variation problem, the variables will be related in the form First find. d = p = =. Substitute and the other nown values into the equation and solve for the unnown value. d = p d =. p 6000 =. p p = p = The sperm whale will eperience a pressure of 00 lb/sq in at a depth of 00 feet below the surface. Eample: The view V from the air is directly proportional to the square root of the altitude A. If the view from horion to horion at an altitude of 6,000 ft is approimately mi, then what is the view from 6,000 FT? Solution: Since this is a direct variation problem, the variables will be related in the form. First find. V = = = = A Substitute and the other nown values into the equation and solve for the unnown value. V V =. =. A 6,000 =. The view from horion to horion is. miles at 6,000 feet.
10 Eample: Atmospheric pressure varies inversely with altitude. If atmospheric pressure is mb at 00 ft above sea level, at what altitude will the atmospheric pressure be 90 mb? Solution: Since this is an inverse variation problem, the variables will be related in the form First find. p = h = 00 = 00 Substitute and the other nown values into the equation and solve for the unnown value. The atmospheric pressure will be 90 mb at, ft. 00 p = h = h 00 h = 90 h =. Eample: The cost of a plastic sewer pipe varies jointly as its diameter and its length. If a 0 ft pipe with a diameter of 6 in costs $.60, then what is the cost of a 6 ft pipe with a diameter of in? Solution: Since this is a joint variation problem, the variables will be related in the form. First find. C = dl.60 = (0.)(0) = =.60.6 Substitute and the other nown values into the equation and solve for the unnown value. It will cost $9. for 6 ft of in diameter pipe. C =.6dl C =.6 6 C = 9.
11 Eample: The weight of a can of baed beans varies jointly with the height and the square of the diameter. If a in high can with a in radius weighs. o, then what is the weight of a in high can with a diameter of 6 in? Solution: This is a joint variation problem; the variables will be related in the form W = hd. = ()() = = First find. Substitute into the variation equation along with other nown values and solve for the unnown value. W W W = 0.7hd = 0.7()(6) = 0.6 The weight of the can will be 0.6 o. Eample: The time required to process a shipment of oysters varies directly with the number of pounds in the shipment and inversely with the number of worers assigned. If,000 lb can be processed by 6 worers in hr, then how long would it tae worers to process,000 lb? Solution: Since this is a combined variation problem, the variables will be related in the form First find. p t = w 000 = 6 = 000 = Substitute into the variation equation along with other nown values and solve for the unnown value p t = w 000(0.06) t = t =. It will tae. hours for worers to process,000 lb of oysters.
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