How do I write the sum of two numbers as the product of their GCF and another sum
2 answers:
Let's assume we were given 2 numbers: 15 and 30. Their sum is:
We want to express it as the product of GCF and another sum.
15 is divisible by: 1, 3, 5, 15
30 is divisible by: 1, 2, 3, 5, 6, 10, 15, 30
The greatest number that appears in 2 series is 15.
In this case sum of two numbers can always be written as:
We want to express it as the product of GCF and another sum.
15 is divisible by: 1, 3, 5, 15
30 is divisible by: 1, 2, 3, 5, 6, 10, 15, 30
The greatest number that appears in 2 series is 15.
In this case sum of two numbers can always be written as:
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The maximum volume of the box is 40√(10/27) cu in.
Here we see that volume is to be maximized
The surface area of the box is 40 sq in
Since the top lid is open, the surface area will be
lb + 2lh + 2bh = 40
Now, the length is equal to the breadth.
Let them be x in
Hence,
x² + 2xh + 2xh = 40
or, 4xh = 40 - x²
or, h = 10/x - x/4
Let f(x) = volume of the box
= lbh
Hence,
f(x) = x²(10/x - x/4)
= 10x - x³/4
differentiating with respect to x and equating it to 0 gives us
f'(x) = 10 - 3x²/4 = 0
or, 3x²/4 = 10
or, x² = 40/3
Hence x will be equal to 2√(10/3)
Now to check whether this value of x will give us the max volume, we will find
f"(2√(10/3))
f"(x) = -3x/2
hence,
f"(2√(10/3)) = -3√(10/3)
Since the above value is negative, volume is maximum for x = 2√(10/3)
Hence volume
= 10 X 2√(10/3) - [2√(10/3)]³/4
= 2√(10/3) [10 - 10/3]
= 2√(10/3) X 20/3
= 40√(10/27) cu in
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