SOLUTION: please help me on my homework. 1.the sum of the numbers is 15. the sum of the squares of the numbers is 113. find the numbers. 2. one number is 6 larger than the other number

Question 108495: please help me on my homework.
1.the sum of the numbers is 15. the sum of the squares of the numbers is 113. find the numbers.
2. one number is 6 larger than the other number. the square of the larger number is 132 greater than the square of the smaller number. what are the numbers?
3. 5 years hence, a boy's age will be 4 times the square of his age 9 years ago. how old is this boy?
4. the perimeter of a square is numerically 4 more than its area. find the length of its side.
5. the width of a rectangle is 9 cm shorter than its diagonal and 7 cm shorter than its length. how long is the diagonal?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
1.the sum of the numbers is 15. the sum of the squares of the numbers is 113. find the numbers.
:
Write the equation for the statements,
"The sum of the numbers is 15"
x + y = 15
or
y = (15 - x);we can use this for substitution
:
"The sum of the squares is 113"
x^2 + y^2 = 113
:
Substitute (15-x) for y in the above equation
x^2 + (15-x)^2 = 113
:
FOIL (15-x)((15-x)
x^2 + 225 - 30x + x^2 = 113
Arrange as a quadratic equation:
x^2 + x^2 - 30x + 225 - 113 = 0
2x^2 - 30x + 112 = 0
:
Simplify, divide by 2
x^2 - 15x + 56 = 0
This factors to:
(x - 8)(x - 7) = 0
:
x = 8 or x = 7
Then
y = 7 or y = 8
:
Check solution: 8^2 + 7^2 = 113
:
:
2. one number is 6 larger than the other number. the square of the larger number is 132 greater than the square of the smaller number. what are the numbers?
:
Let x = "one number"
Then
(x-6) = "the other number
:
Write an equation for what it says:
"the square of the larger number is 132 greater than the square of the smaller number."
x^2 = (x-6)^2 + 132
x^2 = x^2 - 12x + 36 + 132
Arrange the x's on the left
x^2 - x^2 + 12x = 168
12x = 168
x = 168/12
x = 14
You can figure out what the smaller number is; check your solution
:
:
3. 5 years hence, a boy's age will be 4 times the square of his age 9 years ago. how old is this boy?
:
Let x = the boys age now
:
"5 years hence" he will be:
(x + 5)
:
"9 year ago" he was:
(x - 9)
;
Write an equation for the statement:
(x + 5) = 4(x-9)^2
x + 5 = 4(x^2 - 18x + 81)
x + 5 = 4x^2 - 72x + 324
Arrange as a quadratic:
4x^2 - 72x - x + 324 - 5 = 0
4x^2 - 73x + 319 = 0
you probably need the quadratic formula to find x, but it does factor:
(4x - 29)(x - 11) = 0
:
x = 11, it's the integer solution we want here.
:
Check: 16 = 4(11-9)
:
:
4. the perimeter of a square is numerically 4 more than its area. find the length of its side.
:
Let x = the length of the side
Then
4x = the perimeter
and
x^2 = the area
:
The statement says:
4x = x^2 + 4
Arrange as a quadratic equation:
x^2 - 4x + 4 = 0
Factors to:
(x-2)(x-2)
obviously x = +2
:
Check: 4(2) = 2^2 + 4
:
:
5. the width of a rectangle is 9 cm shorter than its diagonal and 7 cm shorter than its length. how long is the diagonal?
:
Let x = the diagonal
Then
(x-9) = width
:
It says the length is 7 cm longer than the width. which is (x-9), therefore
(x-9) + 7 = (x-2) is the length
:
Using pythag we have:
Length^2 + width^2 = diagonal^2
(x-2)^2 + (x-9)^2 = x^2
:
See if you can do this yourself by what was shown above, only one of the solutions will make sense.
If you have any difficulty you can email me.
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