Set the equation equal to zero.
9x+6=0
Subtract 6 from both sides
9x = -6
Divide both sides by 9
x=-6/9
or -2/3
2(5x-3)=0
Distribute
10x-6=0
add 6 to both sides
10x=6
Divide by 10 on both sides
x=6/10 or 4/5
5y+14=0
-14 from both sides
5y=-14
divide both sides by 5
y=-14/5
or -2 and 4/5
Answer:
a) Each corral should be 33⅓ ft long and 25 ft wide
b) The total enclosed area is 1666⅔ ft²
Step-by-step explanation:
I assume that the corrals have identical dimensions and are to be fenced as in the diagram below
Let x = one dimension of a corral
and y = the other dimension
(a) Dimensions to maximize the area
The total length of fencing used is:
4x + 3y = 200
4x = 200 – 3y
x = 50 - ¾y
The area of one corral is A = xy, so the area of the two corrals is
A = 2xy
Substitute the value of x
A = 2(50 - ¾y)y
A = 100 y – (³/₂)y²
This is the equation for a downward-pointing parabola:
A = (-³/₂)y² + 100y
a = -³/₂; b = 100; c = 0
The vertex (maximum) occurs at
y = -b/(2a) = 100 ÷ (2׳/₂) = 100 ÷ 3 = 33⅓ ft
4x + 3y = 100
Substitute the value of y
4x + 3(33⅓) = 200
4x + 100 = 200
4x = 100
x = 25 ft
Each corral should measure 33⅓ ft long and 25 ft wide.
Step 2. Calculate the total enclosed area
The enclosed area is 50 ft long and 33⅓ ft wide.
A = lw = 50 × 100/3 = 5000/3 = 1666⅔ ft²
Well, we could try adding up odd numbers, and look to see when we reach 400. But I'm hoping to find an easier way.
First of all ... I'm not sure this will help, but let's stop and notice it anyway ...
An odd number of odd numbers (like 1, 3, 5) add up to an odd number, but
an even number of odd numbers (like 1,3,5,7) add up to an even number.
So if the sum is going to be exactly 400, then there will have to be an even
number of items in the set.
Now, let's put down an even number of odd numbers to work with,and see
what we can notice about them:
1, 3, 5, 7, 9, 11, 13, 15 .
Number of items in the set . . . 8
Sum of all the items in the set . . . 64
Hmmm. That's interesting. 64 happens to be the square of 8 .
Do you think that might be all there is to it ?
Let's check it out:
Even-numbered lists of odd numbers:
1, 3 Items = 2, Sum = 4
1, 3, 5, 7 Items = 4, Sum = 16
1, 3, 5, 7, 9, 11 Items = 6, Sum = 36
1, 3, 5, 7, 9, 11, 13, 15 . . Items = 8, Sum = 64 .
Amazing ! The sum is always the square of the number of items in the set !
For a sum of 400 ... which just happens to be the square of 20,
we just need the <em><u>first 20 consecutive odd numbers</u></em>.
I slogged through it on my calculator, and it's true.
I never knew this before. It seems to be something valuable
to keep in my tool-box (and cherish always).
Answer:
20 inches²
Step-by-step explanation:
perimeter = 8 + 3 + 5 + 4
= 20 inches²
