A cylinder has a diameter of 24 inches. If its height is half its radlus, what is the volume of the cylinder in cubic Inches?
1 answer:
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Answer:
is the sum of positive integers between
(inclusive) and
(inclusive) that are not multiples of
and not multiples
.
Step-by-step explanation:
For an arithmetic series with:
as the first term,
as the last term, and
as the common difference,
there would be terms, where as the sum would be
.
Positive integers between (inclusive) and
(inclusive) include:
.
The common difference of this arithmetic series is . There would be
terms. The sum of these integers would thus be:
.
Similarly, positive integers between (inclusive) and
(inclusive) that are multiples of
include:
.
The common difference of this arithmetic series is . There would be
terms. The sum of these integers would thus be:
Positive integers between (inclusive) and
(inclusive) that are multiples of
include:
.
The common difference of this arithmetic series is . There would be
terms. The sum of these integers would thus be:
Positive integers between (inclusive) and
(inclusive) that are multiples of
(integers that are both multiples of
and multiples of
) include:
.
The common difference of this arithmetic series is . There would be
terms. The sum of these integers would thus be:
.
The requested sum will be equal to:
- the sum of all integers from
to
,
- minus the sum of all integer multiples of
between
and
, and the sum integer multiples of
between
- plus the sum of all integer multiples of
between
That is:
.
The volume of a cuboid is given by length × width × height
We have:
Volume = 7.6 ft³
Height = 3x - 1
Length = x + 5
Width = x
Substituting these into the formula, we have:
7.6 = (3x - 1) (x + 5) (x)
7.6 = [3x² + 15x - x - 5] (x)
7.6 = [3x² + 14x - 5](x)
7.6 = 3x³ + 14x² - 5x
0 = 3x³ + 14x² - 5x - 7.6
Drawing the graph is one way of finding the solution (refer to the graph below):
We have three solutions (where the curve crosses the x-axis):
x = -4.9
x = -0.6
x = 0.8
Putting these solutions back into the context, since we are looking for the value of x which is part of measurement of length, we cannot have negative value, so we will take the value of x = 0.8 ft
Converting 0.8 ft into inches = 0.8 × 12 inches = 9.6 inches
Answer: x = 9.6 inches
We have:
Volume = 7.6 ft³
Height = 3x - 1
Length = x + 5
Width = x
Substituting these into the formula, we have:
7.6 = (3x - 1) (x + 5) (x)
7.6 = [3x² + 15x - x - 5] (x)
7.6 = [3x² + 14x - 5](x)
7.6 = 3x³ + 14x² - 5x
0 = 3x³ + 14x² - 5x - 7.6
Drawing the graph is one way of finding the solution (refer to the graph below):
We have three solutions (where the curve crosses the x-axis):
x = -4.9
x = -0.6
x = 0.8
Putting these solutions back into the context, since we are looking for the value of x which is part of measurement of length, we cannot have negative value, so we will take the value of x = 0.8 ft
Converting 0.8 ft into inches = 0.8 × 12 inches = 9.6 inches
Answer: x = 9.6 inches