#$&* course mth151
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Given Solution: To say that y is proportional to x is to say that there exists some constant number k such that y = k x. Using the given values of y and x we can determine the value of k: Since y = 9 when x = 12, y = k x becomes 9 = k * 12. Dividing both sides by 12 we obtain 9 / 12 = k. Reducing and reversing sides we therefore obtain k =.75. Now our proportionality reads y = .75 x. Thus when x = 32 we have y = .75 * 32 = 24. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. If y is proportional to the square of x, and y = 8 when x = 12, then what is the value of y when x = 9? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y =1/18*9^2=81/18=4.5 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To say that y is proportional to x is to say that there exists some constant number k such that y = k x^2. Using the given values of y and x we can determine the value of k: Since y = 8 when x = 12, y = k x^2 becomes 8 = k * 12^2, or 8 = 144 k. Dividing both sides by 144 we obtain k = 8 / 144 = 1 / 18. Now our proportionality reads y = 1/18 x^2. Thus when x = 9 we have y = 1/18 * 9^2 = 81 / 18 = 4.5. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q003. If y is inversely proportional to x and if y = 120 when x = 200, when what is the value of y when x = 500? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y=24,000/500 =480 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To say that y is inversely proportional to x is to say that there exists some constant number k such that y = k / x. Using the given values of y and x we can determine the value of k: Since y = 120 when x = 200, y = k / x becomes 120 = k / 200. Multiplying both sides by 200 we obtain k = 120 * 200 = 24,000. Now our proportionality reads y = 24,000 / x. Thus when x = 500 we have y = 24,000 / 500 = 480. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. If y is inversely proportional to the square of x and if y = 8 when x = 12, then what is the value of y when x = 16? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: k = 8 * 144 = 1152. y = 1152 / (16)^2 =4.5 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To say that y is inversely proportional to the square of x is to say that there exists some constant number k such that y = k / x^2. Using the given values of y and x we can determine the value of k: Since y = 8 when x = 12, y = k / x^2 becomes 8 = k / 12^2, or 8 = k / 144. Multiplying both sides by 144 we obtain k = 8 * 144 = 1152. Now our proportionality reads y = 1152 / x^2. Thus when x = 16 we have y = 1152 / (16)^2 = 4.5. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q005. If y is proportional to the square of x and inversely proportional to z, then if y = 40 when x = 10 and z = 4, what is the value of y when x = 20 and z = 12? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y = k x^2 / z 40 = k * 10^2 / 4 k = 1.6 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To say that y is proportional to the square of x and inversely proportional to z is to say that the there exists a constant k such that y = k x^2 / z. Substituting the given values of x, y and z we can evaluate k: y = k x^2 / z becomes 40 = k * 10^2 / 4. Multiplying both sides by 4 / 10^2 we obtain 40 * 4 / 10^2 = k, or k = 1.6. Our proportionality is now y = 1.6 x^2 / z, so that when x = 20 and z = 12 we have y = 1.6 * 20^2 / 12 = 1.6 * 400 / 12 = 53 1/3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q006. If y is proportional to x^2, with y = 9 when x = 2, what is the value of y when x = 17? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q007. If y is inversely proportional to x^3, with y = 9 when x = 7, then what is the value of y when x = 2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ------------------------------------------------ Self-critique Rating:
#$&* course mth151 12/19/2013 029. `query 29
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Given Solution: `a**If we multiply both sides by 6 * 18 we get 6 * 18 * (1/3 ) / 6 = 6 * 18 * (1 / 18) or 18 * 1/3 = 6. Note that the effect here is the same as that of 'cross-multiplying', but it's a good idea to remember that 'cross-multiplying' is really a shortcut way to think of multiplying both sides by the common denominator. Since 18 * 1/3 = 18 / 3 = 6, the equation 18 * 1/3 = 6 is true, which verifies the original equality. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q7.3.20 z/8 = 49/56. Solve this proportionality for z. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: z = 8 * 49 / 56 z = 7 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**Multiply both sides by 8 * 56 to get 8 * 56 * z / 8 = 8 * 56 * 49 / 56. Simplify to get 56 * z = 8 * 49. Divide both sides by 56 to get z = 8 * 49 / 56. Simplify to get z = 7. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q7.3.42 8 oz .45; 16 oz. .49; 50 oz. 1.59`sb Which is the best value per unit for green beans and how did you obtain your result? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 16 oz for .49 best value at 3.06 cents per oz confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** 45 cents / 8 oz = 5.63 cents / oz. 49 cents / 16 oz = 3.06 cents / oz. 159 cents / 50 oz = 3.18 cents / oz. 16 oz for .49 is the best value at 3.06 cents / oz. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q7.3.45 triangles 4/3, 2, x; 4, 6, 3. What is the value of x and how did you use an equation to find it? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x/3=4/3/4 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** the 4/3 corresponds to 4, 2 corresponds to 6, and x corresponds to 3. The ratios of corresponding sides are all equal. So 4/3 / 4 = 2 / 6 = x / 3. Just using x / 3 = 2 / 6 we solve to get x = 1. We would have obtained the same thing if we had used x / 3 = 4/3 / 4. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qIf z = 9 when x = 2/3 and z varies inversely as x, find z when x = 5/4. Show how you set up and used an equation of variation to solve this problem. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 9 = k / ( 2/3) 2/3 * 9 = k so k = 6 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** If z varies inversely as x then z = k / x. Then we have 9 = k / ( 2/3). Multiplying both sides by 2/3 we get 2/3 * 9 = k so k = 6. Thus z = 6 / x. So when x = 5/4 we have z = 6 / (5 /4 ) = 24 / 5 = 4.8. Note that the translations of other types of proportionality encountered in this chapter include: z = k x^2: z varies as square of x. z = k / x^2: z varies inversely as square of x. z = k x: z is proportional to x. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q7.3.72. Illumination is inversely proportional to the square of the distance from the source. Illumination at 4 ft is 75 foot-candles. What is illumination at 9 feet? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: k = 75 * 4^2 = 75 * 16 = 1200 the illumination is 1200 confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**Set up the variation equation I = k / r^2, where I stands for illumination and r for distance (you might have used different letters). This represents the inverse proportionality of illumination with the square of distance. Use I = 75 when r = 4 to get 75 = k / 4^2, which gives you k = 75 * 4^2 = 75 * 16 = 1200. Now rewrite the proportionality with this value of k: I = 1200 / r^2. To get the illumination at distance 9 substitute 9 for r to get I = 1200 / 9^2 = 1200 / 81 = 14.8 approx.. The illumination at distance 9 is about 14.8. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q7.3.66 length inv prop width; L=27 if w=10; w = 18. L = ? Explain how you set up and used a variation equation to obtain the length as a function of width, giving your value of k. Then explain how you used your equation to find the length for width 18 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: L =270/18=15 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**Set up the variation equation L = k / w, which is the inverse proportion. Use L = 27 when w = 10 to get 27 = k / 10, which gives you k = 27 * 10 = 270. Now we know that L = 270 / w. So if w = 18 you get L = 270 / 18 = 15. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q7.3.66 length inv prop width; L=27 if w=10; w = 18. L = ? Explain how you set up and used a variation equation to obtain the length as a function of width, giving your value of k. Then explain how you used your equation to find the length for width 18 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: L =270/18=15 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a**Set up the variation equation L = k / w, which is the inverse proportion. Use L = 27 when w = 10 to get 27 = k / 10, which gives you k = 27 * 10 = 270. Now we know that L = 270 / w. So if w = 18 you get L = 270 / 18 = 15. ** " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!