SOLUTION: First Problem One number is 3 smaller than another number. The square of the larger number is 9 larger than ten times the smaller number. Find the numbers. 2nd Problem S

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Question 250260: First Problem
One number is 3 smaller than another number. The square of the larger number is 9 larger than ten times the smaller number. Find the numbers.
2nd Problem
Suppose that the width of a certain rectangle is 3 inches less than its length. The area is numerically 6 less than twice the perimeter. Find the length and width of the rectangle.
Thank you.
Answer by solver91311(24713) About Me  (Show Source):
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Let represent the smaller number. Then the larger number is

The square of the larger number:

Ten times the smaller number: . Nine more than that is

Equate the two expressions (the sentence has the verb "is" in it)







Solve for . Remember, this is a quadratic, so there will be two roots to the equation. Check them against the parameters of the problem to determine if none, one, or both of the roots are solutions to the problem.

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The area of a rectangle is the length times the width. The perimeter of a rectangle is two times the width plus two times the length.

Let represent the width. Then the length is

The area function in terms of the width is then:



The perimeter function in terms of the width is then:



Twice the perimeter is then



But this is 6 more than the Area, so:



Hence



and



Factor and solve for . Discard the negative root -- width cannot be a negative number. The positive root is the width. Three more than that is the length.


John