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
Algebra.Com
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) (Show Source): You can put this solution on YOUR website!
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.
========================
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:
\ =\ w^2\ +\ 3w)
The perimeter function in terms of the width is then:
\ =\ 2(w\ +\ 3)\ +\ 2w\ =\ 4w\ +\ 6)
Twice the perimeter is then
\ =\ 8w\ +\ 12)
But this is 6 more than the Area, so:
\ =\ 2P(w)\ -\ 6)
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
RELATED QUESTIONS
One number is 3 smaller than another number. The square of the larger number is 9 larger... (answered by solver91311)
one number is 3 smaller than another number. the square of the larger number is 9 larger... (answered by checkley79)
One number is six larger than another number. The square of the larger number is 132... (answered by jhunjiro)
One number is three times another number. Seven times the smaller number is 3 more than... (answered by lwsshak3)
one number is 15 more than another. the sum of twice the larger number and 3 times the... (answered by dabanfield)
one number is 3 more than another. the smaller number decreased by four times the larger... (answered by ankor@dixie-net.com)
one number is 3 more than another. the smaller number decreased by 4 times the larger... (answered by edjones)
One number seven more than another number if the sum of twice the smaller number +3... (answered by hkwu)
One number is 4 more than another. If 3 times the smaller number is equal to twice the... (answered by Fombitz)