The dimensions of a right circular cylindrical can that holds 5832 cubic cm and uses the least amount of material are a radius of approximately 10.614 cm and a height of approximately 16.526 cm.
Explanation:The question asks for the dimensions of a right circular cylindrical can that holds 5832 cubic cm of liquid and is designed to use the least amount of material. This scenario fits into the realm of mathematical optimization. Here, we’re being asked to identify dimensions that minimize the surface area of the cylinder while holding a fixed volume. This kind of problem is typically addressed using calculus.
The volume V of a cylinder is given by the formula V = πr²h where r is the cylinder radius and h is the cylinder height. The surface area A of a cylinder closed at both ends is given by A = 2πrh + 2πr².
From the volume equation, we can express h in terms of V and r, h = V/πr². Substituting this into the surface area equation yields: A = 2πr(V/πr²) + 2πr² = 2V/r + 2πr².
To find the radius that minimizes the surface area, we would set the derivative of the surface area with respect to r equal to zero and solve for r. After these calculations, we find that r = (V/(2π))^(1/3). For V = 5832 cubic cm, this yields r ≈ 10.614 cm.
Substitute r ≈ 10.614 cm back into the equation h = V/πr² to find h ≈ 16.526 cm. So, the dimensions of the can that minimizes the material used are radius 10.614 cm and height 16.526 cm.
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To find the dimensions of the cylindrical can that holds a specific volume and is made from the least material, utilize the volume and surface area formulas for a cylinder. Using calculus, find the radius that minimizes the surface area, then find the corresponding height.
Explanation:This problem is a classic calculus optimization problem. The volume of the cylindrical can is given by the formula V = πr²h, where r is the radius and h is the height of the cylinder. To minimize the amount of material used in constructing the can, we need to minimize the surface area, given by the formula [tex]A = 2\pi rh + 2\pi r^2[/tex] (which includes both top and bottom of the can).
From the volume formula, we can express h as h = V/πr² = 5832/πr². Substituting this value into the surface area equation, we get A = [tex]2\pi r(5832/\pi r^2) + 2\pi r^2 = 11664/r + 2\pi r^2.[/tex] Differentiating this equation with respect to r and setting the derivative equal to zero, will provide the radius that minimizes the material used. The height can be obtained by substituting this radius value back into the equation h = 5832/πr².
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You have a rectangular vegetable garden in your backyard that measures 15 feet by 10 feet. You want to double the area of the garden by adding the same distance x to the length and width of the garden. Find the value of x and the new dimensions of the garden.
To double the area of the rectangular vegetable garden, we need to add the same distance, x, to the length and width. The value of x is 10 feet, and the new dimensions of the garden are 25 feet by 20 feet.
Explanation:To double the area of the garden, we need to add the same distance, x, to both the length and width of the garden. The original dimensions of the garden are 15 feet by 10 feet, with an area of 150 square feet. To double the area, we set up the equation 2(15+x)(10+x) = 2(15)(10). Solving this equation gives us the value of x and the new dimensions of the garden.
Expanding the equation, we have 30x + x² + 300 = 300. This simplifies to x² + 30x - 300 = 0. Factoring this quadratic equation, we get (x - 10)(x + 30) = 0. So x = -30 or x = 10. We discard the negative value since distance cannot be negative, so x = 10 feet.
The new dimensions of the garden are 15+10 = 25 feet and 10+10 = 20 feet.
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A rectangle has one side on the xx-axis and two vertices on the curve y=74+x2. y=74+x2. find the vertices of the rectangle with maximum area.
The rectangle with maximum area does not exist as the curve y = 74 + x^2 does not intersect the x-axis.
Explanation:To find the vertices of the rectangle with maximum area, we need to find the points of intersection between the curve y = 74 + x^2 and the x-axis.
Setting y = 0, we can solve the equation 74 + x^2 = 0 to find the x-coordinates of the vertices.
Since the curve is a parabola facing upwards, there are no points of intersection with the x-axis.
Therefore, the rectangle does not exist.
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The length of a rectangle is 4.2 cm more than 2 times the width. If the perimeter of the rectangle is 42.6 cm, what are its dimensions?
Simplify this problem: X3 / X.
The linear pair postulate states that if two angles form a linear pair, then they are __________ angles.
Answer:
supplementary
Step-by-step explanation:
Joel earns $1,700 per month. If he spends $425 on rent each month, what percent of his income does he spend on rent? A. 20% B. 331/3% C. 40% D. 25%
Given: M is the midpoint of LN; LM =8 prove:LN=16
The proof of LN= 16 is shown below:
What is midpoint?The midpoint formula is applied when one is required to find the exact center point between two defined points. So for a line segment, use this formula to calculate the point that bisects a line segment defined by the two points.
Given:
LM= 8
and M is the midpoint of LN
So, the line LN will be break into two equal parts LM and MN
then we can write
LM= MN= 8
Now,
LN= LM+ MN
LN= 8+8
L=16
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Is 6/36 and 1/6 proportional?
W far has a person traveled who ran at a rate of 5.5 miles per hour for of an hour?
Answer:
4.125 hope this helps!
Step-by-step explanation:
I know u meant 3/4 of an hour
An athlete eats 45g of protein per day while training. How much protein will she eat during 15 days of training?
Answer:
Step-by-step explanation:
Kendall is stacking boxes that are 7.5inches tall. Explain how to find the height (in inches)
Final answer:
To find the height of stacked boxes, multiply the height of one box (7.5 inches) by the total number of boxes stacked.
Explanation:
Kendall is stacking boxes that are each 7.5 inches tall. To find the total height in inches of the stacked boxes, you would need to multiply the height of one box by the total number of boxes stacked. For example, if Kendall stacks 10 boxes, the calculation would be 7.5 inches x 10, which equals 75 inches. Therefore, the total height of the 10 stacked boxes would be 75 inches.
Write the solution to the given inequality in interval notation.
[-3,-1)
(-3,-1]
(-∞,-3) U [-1,∞)
(-∞,-3] U (-1,∞)
Answer:
D. [tex](-\infty,-3]\text{U}(-1,\infty)[/tex]
Step-by-step explanation:
We have been given solution of an inequality on number line. We are asked to write the solution to the given inequality in interval notation.
We can see that all values of x less than or equal [tex]-3[/tex] and greater than [tex]-1[/tex] are solution for our given inequality.
We have a solid dot at [tex]-3[/tex], so [tex]-3[/tex] is a solution of inequality.
We have a open dot at [tex]-1[/tex], so [tex]-1[/tex] is not a solution of inequality.
Upon combining both solutions, we will get:
[tex](-\infty,-3]\text{U}(-1,\infty)[/tex]
Therefore, the solution for the inequality is [tex](-\infty,-3]\text{U}(-1,\infty)[/tex].
Which two graphs are graphs of polynomial functions?
It’s B and D for Plato users
Which statements are true the ordered pair (1, 2) and the system of equations?
y=−2x+4
7x−2y=3
Select each correct answer.
When (1, 2) is substituted into the first equation, the equation is true.
When (1, 2) is substituted into the first equation, the equation is false.
When (1, 2) is substituted into the second equation, the equation is true.
When (1, 2) is substituted into the second equation, the equation is false.
The ordered pair (1, 2) is not a solution to the system of linear equations.
The ordered pair (1, 2) is a solution to the system of linear equations.
The solution to -5m = -20 is negative.
True or false
False
a negative number times a negative number would have a positive answer
The radius of a circle is 10 cm. What is the volume?
The coordinates of point D are (7, 4) and the coordinates of point E are (1, −3) .
What is the slope of the line that is perpendicular to DE¯¯¯¯¯?
Enter your answer as a simplified fraction in the box.
jasmine has $15 dollars and save $2.50 every month . radha has $0 and save $3.50 every month. jasmines savings after x months can be represented by the function f(x)=2.5x+15. radahs savings after x months can be represented by the function g(x)=3.5x after how any moths will they both have the same amount in savings? find the value of x which f(x)=g(x)
2.5x+15 = 3.5x
subtract 2.5x from each side
15 = x
they will have the same amount after 15 months
Solve the equation. How old is Sam?
Sam's age doubled plus 5 is the same as Sam's age tripled minus 1.
Question 5 options:
Sam is 6yrs old.
Sam is 4 yrs old.
Sam is 12 yrs old.
Sam is 5 yrs old.
\\
@Mochimin
What could the answer be?
HELP PLEASE ASAP!
BRAINLIESTTTTTTTTTTTTT
When the federal reserve wants more money to be printed, which agency will be in charge of printing it?
Answer:
The U.S. Treasury.
Step-by-step explanation:
Of the birds in Monica's pet shop, 1/5 are pigeons. Of the pigeons, 1/3 are white. Look at each expression. Can it be used to find the fraction of birds in Monica's pet shop that are white pigeons?
1/15.
The expression that can be used to find the fraction of birds in Monica's pet shop that are white pigeons is:
1/5 x 1/3 = 1/15
This calculation is derived by multiplying the fraction of pigeons in the pet shop (1/5) by the fraction of white pigeons among pigeons (1/3).
The area of a rectangular piece of sheet metal is 216 ft. the width of the sheet is 6 ft less than the length what are the dimensions of th epiece of metal
Using algebra, the dimensions of the rectangular sheet metal with an area of 216 sq ft and width being 6 ft less than the length are found to be 18 ft by 12 ft.
The area of a rectangular piece of sheet metal is given as 216 square feet, and the width is stated to be 6 feet less than the length. Use algebra to find the length (l) and the width (w) of the sheet. Since the width is 6 ft less than the length, we can express this relationship as: w=l- 6.
The area of a rectangle is calculated by the formula Area = length imes width, so we can set up the equation 216 = l imes (l - 6). Solving the quadratic equation l^2 - 6l - 216 = 0 we find that the possible values for l are 18 ft (acceptable since length is positive) and -12 ft (which is not physically meaningful for this context), leaving us with the length of the piece of metal being 18 feet. Consequently, the width would be l - 6, which is 18 - 6 = 12 ft.
Therefore, the dimensions of the piece of metal are 18 feet in length and 12 feet in width.
Anna has 10 to the 2nd power quarters. Jazmin has 10 to the second power dimes Who has more money, Anna or Jasmine? How much more? Explain
This question is based on the concept of logical reasoning.Therefore,they both have the same amounts of coins, the one with 100 quarters has more money.Thus, Anna has more money than Jasmine.
Given:
Anna has 10 to the 2nd power quarters. Jazmin has 10 to the second power dimes.
We need to calculate the who has more money, Anna or Jasmine.
According to the question,
Anna has 10 to the 2nd power quarters.
In mathematically expressed as,
[tex]\bold{10^2 = 100}[/tex] quarters
Now, Jazmin has 10 to the second power dimes.
In mathematically expressed as,
[tex]\bold{10^2 = 100}[/tex] dimes
So,they both have 100 coins. But, we know that quarter is 25 cents, while a dime is only 10 cents.
Anna has 100 quarters = [tex]\bold{100 \times 25 = 2500}[/tex]. That is 25 dollars.
Jasmine has 100 dimes = [tex]\bold{100 \times 10 = 1000}[/tex]. This is 10 dollars.
Therefore,they both have the same amounts of coins, the one with 100 quarters has more money.Thus, Anna has more money than Jasmine.
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A movie rental website charges $5.00 per month for membership and $1.25 per movie.How many movies did Andrew rent this month if this month's bill was $16.25?
PLEASE HELP ME, I NEED IT.
Which of the following is the correct graph of the compound inequality 4p + 1 > −7 or 6p + 3 < 33?
A. a number line with open circles at negative 2 and 5 with shading in between.
B. number line with open dot at negative 2 with shading to the left and an open dot at 5 with shading to the right
C. number line with shading everywhere.
D. number line with open dot at negative 2 and a closed dot at 5 with shading in between.
The answer is B :)
I hope this helped
Factor the four-term polynomial.
xz + x + yz + y
A. (z - y)(x + 1)
B.(x - y)(z + 1)
C.(x + y)(z + 1)
A master electrician earns $62 per hour. His apprentice earns $40 per hour. The master electrician works 3 hours more than the apprentice. If together they are paid $492, how much does the master electrician earn?
A) $292
B) $312
C) $332
D) $372
Answer:
The answer is D)$372
Step-by-step explanation:
Rules for Plotting Points:
1) Start at the origin: _________
2) If your x-value is ___________, move right; if it is ___________, move left;
if it is _____, don’t move.
3) If your y-value is ___________, move up; if it is ___________, move down;
if it is _____, don’t move.