Provides data on sleep deprivation rates of Californians and Oregonians. The proportion of California residents who reported ins
1 answer:
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Answer:
Step-by-step explanation:
<u>Optimizing With Derivatives </u>
The procedure to optimize a function (find its maximum or minimum) consists in :
- Produce a function which depends on only one variable
- Compute the first derivative and set it equal to 0
- Find the values for the variable, called critical points
- Compute the second derivative
- Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum
We know a cylinder has a volume of 4 . The volume of a cylinder is given by
Equating it to 4
Let's solve for h
A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is
Replacing the formula of h
Simplifying
We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero
Rearranging
Solving for r
Computing h
We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative
We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.
The minimum area is
Answer:
This is because modulus number will be always positive
Hope this is correct and helpful
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Answer:
- As the ratio for all the points on the equation y = x is same. Hence, the equation y = x REPRESENTS a proportional relationship.
- As the ratio for all the points on the equation y = x+2 is not same. Hence, the equation y = x + 2 DOES NOT REPRESENT a proportional relationship.
Step-by-step explanation:
<u><em>Solving equation A: y = x </em></u>
Let us consider the given equation A:
y = x
- Putting x = 1 in y = x
y = x
y = 1 ∵ x = 1
Hence, (1, 1) is the ordered pair of y = x
- Putting x = 2 in y = x
y = x
y = 2 ∵ x = 2
Hence, (2, 2) is the ordered pair of y = x
- Putting x = 3 in y = x
y = x
y = 3 ∵ x = 3
Hence, (3, 3) is the ordered pair of y = x
- Putting x = 4 in y = x
y = x
y = 4 ∵ x = 4
Hence, (4, 4) is the ordered pair of y = x
- Putting x = 5 in y = x
y = x
y = 5 ∵ x = 5
Hence, (5, 5) is the ordered pair of y = x
Lets us consider all the ordered pairs i.e. (1, 1), (2, 2), (3, 3), (4, 4) and (5, 5) to make a table for y = x.
<em>y x</em>
1 1
2 2
3 3
4 4
5 5
As from the table, lets take the ratio of every point i.e. y/x
- For (1, 1), the ratio will be y/x ⇒ 1/1 = 1
- For (2, 2), the ratio will be y/x ⇒ 2/2 = 1
- For (3, 3), the ratio will be y/x ⇒ 3/3 = 1
- For (4, 4), the ratio will be y/x ⇒ 4/4 = 1
- For (5, 5), the ratio will be y/x ⇒ 5/5 = 1
Hence, the ratio for all the points on the equation y = x is same. Hence, the equation y = x REPRESENTS a proportional relationship. Please also check the graph in attached<em> figure a.</em>
<u><em>Solving equation A: y = x +2</em></u>
Let us consider the given equation A:
y = x + 2
- Putting x = 1 in y = x + 2
y = x + 2
y = 1 + 2 ⇒ 3 ∵ x = 1
Hence, (1, 3) is the ordered pair of y = x + 2
- Putting x = 2 in y = x + 2
y = x + 2
y = 2 +2 ⇒ 4 ∵ x = 2
Hence, (2, 4) is the ordered pair of y = x + 2
- Putting x = 3 in y = x + 2
y = x + 2
y = 3 + 2 ⇒ 5 ∵ x = 3
Hence, (3, 5) is the ordered pair of y = x + 2
- Putting x = 4 in y = x + 2
y = x + 2
y = 4 + 2 ⇒ 6 ∵ x = 4
Hence, (4, 6) is the ordered pair of y = x + 2
- Putting x = 5 in y = x + 2
y = x + 2
y = 5 +2 ⇒ 7 ∵ x = 5
Hence, (5, 7) is the ordered pair of y = x + 2
Lets us consider all the ordered pairs i.e. (1, 3), (2, 4), (3, 5), (4, 6) and (5, 7) to make a table for y = x + 2.
<em>y x + 2</em>
1 3
2 4
3 5
4 6
5 7
As from the table, lets take the ratio of every point i.e. y/x
- For (1, 3), the ratio will be y/x ⇒ 3/1 = 3
- For (2, 4), the ratio will be y/x ⇒ 4/2 = 2
- For (3, 5), the ratio will be y/x ⇒ 5/3 = 5/3
- For (4, 6), the ratio will be y/x ⇒ 6/4 = 3/2
- For (5, 7), the ratio will be y/x ⇒ 7/5 = 7/5
Hence, the ratio for all the points on the equation y = x+2 is not same. Hence, the equation y = x + 2 DOES NOT REPRESENT a proportional relationship. Please also check the graph in attached<em> figure a.</em>
<em>Keywords: equation, graph</em>
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