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1 Let t represent the number of hours spent running from home to 5 school Then t 60 represents the number of hours spent running from school to home The distance to school is given by d 10t, 5 and the distance home is given by d 9 t 60   5 10t 9 t 60   1 10t 9 t 12 10t 9t 10t 9t 9t 9 12 3 4
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9t 3 t 4 The distance between home and school is 10t 10 3 15 71 4 2 2 miles 2 Let t represent the number of hours Toni spent driving to work Then t 18 represents the number of hours driving home The dis60 tance from home to work is given by d 30t, and the distance from work to home is given by d 45 t 18 60   18 30t 45 t 60   3 30t 45 t 10 30t 45t 30t 45t 45t 45t 135 10 27 2
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15t t t 27 2 1 27 15 2
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9 10 9 The distance from Toni s home and work is 30t 30 10 27 miles 3 Let t represent the number of hours Leo spent walking his grandson to school Then t 16 represents the number of hours Leo 60 spent walking home The distance from home to school is given by d 3t, and the distance from school to home is given by d 5 t 16 60   16 3t 5 t 60   4 3t 5 t 15 20 15 4 3t 5t 3 5t 5t 4 2t 3 1 4 t 2 3 2 t 3 3t 5t The distance from home to school is 3t 3 2 2 miles 3 In the above examples and practice problems, the number for t was substituted in the rst distance equation to get d It does not matter which equation you used to nd d, you should get the same value If you do not, then you have made an error somewhere
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Algebra problems involving geometric gures are very common In algebra, you normally deal with rectangles, triangles, and circles On occasion, you will be asked to solve problems involving other shapes like right circular cylinders and right circular cones If you master solving the more common types of geometric problems, you will nd that the more exotic shapes are just as easy In many of these problems, you will have several unknowns which you must reduce to one unknown In the problems above, you reduced a problem of three unknowns to one unknown by relating one quantity to another (the time on one direction related to the time on the return trip) and by setting the equal distances equal to each other We will use similar techniques here
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Example
A rectangle is 11 times as long as it is wide The perimeter is 100 cm2 2 Find the dimensions of the rectangle The formula for the perimeter of a rectangle is given by P 2l 2w We are told the perimeter is 100, so the equation now becomes 100 2l 2w We are also told that the length is 11 times the width, 2 so l 1:5w We can substitute this l into the equation: 100 2l 2w 2 1:5w 2w We have reduced an equation with three unknowns to one with a single unknown 100 2 1:5w 2w 100 3w 2w 100 5w 100 w 5 20 w The width is 20 cm and the length is 1:5w 1:5 20 30 cm
Practice
1 A box s width is 2 its length The perimeter of the box is 40 inches 3 What are the box s length and width 2 A rectangular yard is twice as long as it is wide The perimeter is 120 feet What are the yard s dimensions
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