Answer:
The cost of the can is minimal when r=2 m and h=5 m
Step-by-step explanation:
This is an optimization problem which will be solved by the use of derivatives
We must find an expression for the total cost of the cylindrical can and then find its dimensions to make the cost minimal
Let's picture a cylinder of radius r and height h. Its volume is computed as
[tex]V=\pi r^2h[/tex]
To make the lids we need two circle-shaped pieces of a material which costs $10 per square meter
The area of each lid is
[tex]A_l=\pi r^2[/tex]
The cost of both lids will be
[tex]C_l=(2)(10)(\pi r^2)=20\pi r^2[/tex]
The lateral side of the cylinder can be constructed with a rectangle of material which costs $8 per square meter
The rectangle has a height of h and a width equal to the length of the circumference
[tex]A_s=2\pi rh[/tex]
The cost of the side of the cylinder will be
[tex]C_s=16\pi rh[/tex]
The total cost of the can is
[tex]C=20\pi r^2+16\pi rh[/tex]
We know the volume of the can is [tex]20\pi[/tex] cubic meters
[tex]V=\pi r^2h=20\pi[/tex]
Isolating h we have
[tex]h=\frac{20}{r^2}[/tex]
Using this value into the total cost
[tex]C=20\pi r^2+16\pi r(\frac{20}{r^2})[/tex]
[tex]C=20\pi r^2+\frac{320\pi}{r}[/tex]
Differentiating with respect to r
[tex]C'=40\pi r-\frac{320\pi}{r^2}[/tex]
To find the critical point, we must set C'=0
[tex]40\pi r-\frac{320\pi}{r^2}=0[/tex]
Operating
[tex]40\pi r^3-320\pi=0[/tex]
Solving for r
[tex]r=\sqrt[3]{\frac{320\pi}{40\pi}}=\sqrt[3]{8}[/tex]
r=2
And since
[tex]h=\frac{20}{r^2}=\frac{20}{2^2}[/tex]
h=5
Differentiating again we find
[tex]C''=40\pi +\frac{640\pi}{r^3}[/tex]
Which is always positive for r positive. It makes the critical point found a minimum
The cost of the can is minimal when r=2 m and h=5 m
A line passes through the point (9. -9) and has a slope of -4/3.
Write an equation in point-slope form for this line.
Answer:
[tex]y = \frac{ - 4}{3} x + 3[/tex]
Step-by-step explanation:
[tex]using \: point \: slope \: form \\ y - y1 = m(x - x1) \\ y + 9= \frac{ - 4}{3} (x - 9) \\ y + 9= \frac{ - 4}{3}x + 12\\ y = \frac{ - 4}{3} x + 3[/tex]
The equation of the line in point-slope form passing through (9, -9) with a slope of -4/3 is y + 9 = -4/3 * (x - 9).
Explanation:
The point-slope form of a line's equation is expressed as y - y1 = m(x - x1). In this equation, 'm' represents the line's slope and (x1, y1) defines a particular point through which the line passes. In this scenario, the slope is given as '-4/3' and the point (9, -9). Consequently, by substituting these parameters into the point-slope form, we obtain the equation y - (-9) = (-4/3)(x - 9) or y + 9 = -4/3 * (x - 9).
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x/-9=3 what is x add explanation
Answer:
x=-27
Step-by-step explanation:
x/-9=3
x=3*-9=-27
Which expression uses the associative property to make it easier to evaluate? (50points)
The correct answer is option 4: [tex](8.\frac{1}{4}).\frac{3}{5}[/tex]
Step-by-step explanation:
Given expression is:
[tex]8.(\frac{1}{4}.\frac{3}{5})[/tex]
The associative property states that for any three terms that are being multiplied, different order of multiplication (i.e. which two terms are multiplied first) will give the same answer.
Let a,b and c be the three terms
then
[tex]a.(b.c) = (a.b).c[/tex]
So applying the associative property on given expression
[tex]8.(\frac{1}{4}.\frac{3}{5}) = (8.\frac{1}{4}).\frac{3}{5}[/tex]
Hence,
The correct answer is option 4: [tex](8.\frac{1}{4}).\frac{3}{5}[/tex]
Keywords: Associative property, Properties
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Write an equation in point-slope form for the line passing through the given point with the given slope. (5,2) m=3
Answer:
y - 2 = 3(x - 5)
Step-by-step explanation:
The equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b) a point on the line
Here m = 3 and (a, b) = (5, 2), thus
y - 2 = 3(x - 5) ← equation of line
Answer:
y - 2 = 3(x - 5)
Step-by-step explanation:
Start with the general point-slope equation:
y - k = m(x - h)
Now substitute 2 for k, 5 for h and 3 for m:
y - 2 = 3(x - 5)
Two factory plants are making TV panels. Yesterday, Plant A produced 16,000 panels. Five percent of the panels from
Plant A and 2% of the panels from Plant B were defective.
How many panels did
Plant B produce, if the overall percentage
of defective panels from the two plants was 4%?
Please help
Plant B produced 8000 panels.
Let's denote the number of panels produced by Plant B as B .
The total number of defective panels from Plant A is [tex]\( 0.05 \times 16000 = 800 \)[/tex] , and the total number of defective panels from Plant B is [tex]\( 0.02 \times B \)[/tex].
The total number of panels from both plants is 16000 + B .
The overall percentage of defective panels is given as 4%, so:
[tex]\[ \frac{800 + 0.02 \times B}{16000 + B} = 0.04 \][/tex]
Now, let's solve for B :
[tex]\[800 + 0.02 \times B = 0.04 \times (16000 + B) \][/tex]
[tex]\[ 800 + 0.02 \times B = 640 + 0.04 \times B \][/tex]
Subtracting [tex]\( 0.02 \times B \)[/tex] from both sides:
[tex]\[ 800 = 640 + 0.02 \times B \][/tex]
Subtracting 640 from both sides:
[tex]\[ 160 = 0.02 \times B \][/tex]
Now, divide by 0.02 to find B :
[tex]\[ B = \frac{160}{0.02} \][/tex]
B = 8000
So, Plant B produced 8000 panels.
The number of panels produced by Plant B is 8,000.
The number of panels produced by Plant A = 16,000
The number of defective panels from A = 5%
Now, 5% of 16,000 =
So, out of 16,000 panels produced by pant A , 800 are defective. .. (1)
Let us assume number of panels produced by Plant B = m
The number of defective panels from B = 2%
Now, 2% of m
So, out of total m panels produced by pant B , 0.02 m are defective. .. (2)
Now, total panels produced over all = Panels by A + Panels by B
= 16,000 + m
The percentage of defective panels over all = 4%
Now, 4% of (16,000 + m)
Also, the total number of defective panels = Defective from A + Defective from B
⇒(0.04)(16,000 + m) = 800 + 0.02 m from (1) and (2)
0.02 m = 160
⇒ m = 160 /0.02 = 8,000
or, m = 8,000
Hence, Number of panels produced by Plant B is 8,000.
Which of the following describes how to translate the graph y = Ixl to obtain the graph of y = 1x - 11 - 1?
1 unit left and 1 unit down
1 unit left and 1 unit up
1 unit right and 1 unit down
1 unit right and 1 unit up
A statement that best describes how to translate the graph y = Ixl to obtain the graph of y = |x - 1| - 1 is: C. 1 unit right and 1 unit down.
In Mathematics, the vertex form of the equation for an absolute value function can be modeled by the following:
y = a|x - h| + k.
Where:
h and k are the vertex of the graph.a is a numerical constant.By critically observing the equation of the transformed absolute value function, we can logically deduce that the parent absolute value function y = |x| was horizontally shifted to the right by 1 unit and then vertically shifted 1 unit down as follows;
y = a|x - h| + k.
y = |x - 1| - 1
Complete Question:
Which of the following describes how to translate the graph y = Ixl to obtain the graph of y = |x - 1| - 1?
1 unit left and 1 unit down
1 unit left and 1 unit up
1 unit right and 1 unit down
1 unit right and 1 unit up
Discrimination of Y= -4x^2+ 2x+1
Answer:
20
Step-by-step explanation:
I assume you mean discriminant.
For a quadratic y = ax² + bx + c, the discriminant is b² − 4ac.
Here, a = -4, b = 2, and c = 1. So the discriminant is:
(2)² − 4(-4)(1) = 20
Use an example to describe the multiplicative relationship between two equivalent ratios
Answer:
Step-by-step explanation:
5/10 and 1/2 are both equivalent ratios resulting in half of themselves you can multiply these by any numbers but they will stay as half ratios such as 1:2 x 6 it would be 6:12 divide it by 6 and you have a half
Equivalent ratios have a multiplicative relationship, where multiplying both terms of a ratio by the same number results in another equivalent ratio. For example, multiplying the ratio 1:2 by 2 gives the equivalent ratio 2:4, showing that the ratios are proportional to each other.
Explanation:Understanding Equivalent Ratios through Multiplication
We can explore the concept of equivalent ratios by looking at their multiplicative relationship. For instance, let's consider the ratios 1:2 and 2:4. These two ratios are equivalent because if we multiply both terms of the first ratio by 2, we get the second ratio. Mathematically, this can be expressed as 1x2/2x2 = 2/4, which is the same as the second ratio. Hence, the multiplication of both the numerator and the denominator of the first ratio by the same number (in this case, 2) maintains the ratio's equivalence to the second ratio.
This concept is crucial when dealing with proportions, unit rates, and unit scales. For example, if a map uses a unit scale of 1 inch = 100 feet, doubling the distance on the map to 2 inches would also double the actual distance to 200 feet, showing a direct multiplicative relationship between the two equivalent ratios.
Understanding how to use multiplication to find equivalent ratios is fundamental in solving problems that involve scaling, such as converting units or working with maps and models.
Deb has 26 ounces of shredded cheese. She is making 4 small loaves of garlic cheese bread . How many ounces of cheese can deb use in each loaf of bread. Explain your reasoning
Deb can use 6.5 ounces of cheese per loaf of bread. The calculation is made by dividing the total amount of cheese by the number of loaves.
Explanation:Deb has 26 ounces of shredded cheese and she is making 4 small loaves of garlic cheese bread. To equally distribute the cheese across each loaf, we perform division.
We divide the total amount of cheese by the number of loaves: Thus, 26 ounces divided by 4 loaves equals 6.5 ounces of cheese per loaf.
The reasoning here is simple: we are equally distributing the cheese across each loaf, so we divide the total amount by the number of loaves. This ensures each loaf gets the same amount of cheese.
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Change the expression into a fraction:
(3x/2y+3) + (x^2+3x/4xy−3−2y+6x)
Answer:
[tex] \frac{3x}{2y + 3} + \frac{ {x}^{2} + 3x}{4xy - 3 - 2y+6x} [/tex]
combine like terms afterwards .. for the polynomial
Given the function f(x) = 2x+ 8x - 1.
f(0) =
Answer:
-1
Step-by-step explanation:
if x = 0...
2(0) + 8(0) - 1 =
0 + 0 - 1 =
0 - 1 =
-1
Factor the trinomial 8x^2-12x-8
Answer:
[tex](x-2)(8x+4)[/tex]
Step-by-step explanation:
we have
[tex]8x^{2} -12x-8[/tex]
Equate to zero and find the roots of the quadratic equation
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex]
is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]8x^{2} -12x-8=0[/tex]
so
[tex]a=8\\b=-12\\c=-8[/tex]
substitute in the formula
[tex]x=\frac{-(-12)(+/-)\sqrt{-12^{2}-4(8)(-8)}} {2(8)}[/tex]
[tex]x=\frac{12(+/-)\sqrt{400}} {16}[/tex]
[tex]x=\frac{12(+/-)20} {16}[/tex]
[tex]x_1=\frac{12(+)20} {16}=2[/tex]
[tex]x_2=\frac{12(-)20} {16}=-\frac{1}{2}[/tex]
therefore
[tex]8x^{2} -12x-8=8(x-2)(x+\frac{1}{2})=(x-2)(8x+4)[/tex]
help i don’t understand
Answer: Percy, None, 288
Step-by-step explanation:
Divide to find the rate.
Divide 33/2= about 16. 43/3= about 14.
Percy is faster.
Second problem: 50/12=4
120/30=4
No one, they grade the same amount.
Third:
72/2= 36 per hour
36x8= 288
What is slope of the line that passes through the point (0,7)and(8,7)
Answer:
0
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(7-7)/(8-0)
m=0/8=0
Answer:
0
Step-by-step explanation:
What whole number divided by a decimal can equal 8
Answer: 20 ÷ 2.5 = 8
Step-by-step explanation:
Answer:
4
Step-by-step explanation:
4 divided by 0.5 equals 8.
Jenny drew a figure in art class. Does it have rotational symmetry? If yes, what is the angle of rotation.
Answer:
60°
Step-by-step explanation:
Jenny drew a figure in art school and the figure is shown in the attached diagram.
The diagram is a regular hexagon and it has rotational symmetry.
Two of its sides is placed parallel to the x-axis.
Therefore, by rotating 60° the adjacent sides will become parallel to the x-axis and the figure remains the same and symmetric with respect to the coordinate axes. (Answer)
Answer:
60°Step-by-step explanation:
The fiven figure is a hexagon, which is a regular polygon with six equal sides.
Now, the symmetry of a hexagon is based on a 60° rotation, because each symmetrical portion of the figure enclosed a 60° angle.
Therefore, the right answer is 60°.
2. Two friends work similar jobs at different companies. One gets paid $1458.00
semi-monthly and the other one gets paid $1346.00 bi-weekly. Which one has a
higher annual salary?
Answer:
If by bi-weekly you mean once every two weeks, then the friend who is payed bi-weekly has a higher annual salary. The same for if he is payed twice a week.
Step-by-step explanation:
1458 x 24 = 34992
1346 x 26 =34996
What is two minus Parentheses eight -5×6 parentheses?
Answer: 24
Step-by-step explanation:
2-(8-5x6) = 24
hi answer please thank you
Answer:
1/4
Step-by-step explanation:
You go up 2 and right 8, which translates to 2/8.
Simplify that to 1/4
Plzzzz help ASAP!!!
Answer:
(4y +8 )÷6 -12
Variable = y
Sum = 4y +8
Quotient = 6
coefficient =4
Brenda has $500 in her bank account Every week, she withdraws $40 for expeses. Without making any deposits, how many weeks can she withdraw this money if she wants to maintain a balance of a least $200
Answer:
7 weeks
Step-by-step explanation:
500-200=300. 300 divided by 40 is 7.5 so 7 weeks
Find the slope of the line passing through (-10,5) , (5,5)?
Answer:
0Step-by-step explanation:
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We have two points (-10, 5) and (5, 5).
Substitute:
[tex]m=\dfrac{5-5}{5-(-10)}=\dfrac{0}{15}=0[/tex]
The slope is equal 0. Therefore given line is a horizontal line.
The equation: y = 5- © MP.6 Be Precise Anna's family had a
party in a restaurant. Their food bill was
$214.58. The tip for the servers was found
by multiplying the food bill by 0.18. What
was the cost of the food bill and tip before
ales tax? Round to the nearest cent.
MP.1 Make Sense and Persevere
ghan's vegetable garden is pictured
w. What is its area?
We can see here that the cost of the food bill and tip before sales tax was approximately $253.20.
To find the cost of the food bill and tip before sales tax, you first need to calculate the tip amount and then add it to the food bill.
Given:
Food bill = $214.58
Tip rate = 0.18 (18% expressed as a decimal)
First, calculate the tip amount:
Tip = Food bill × Tip rate
Tip = $214.58 × 0.18
Tip = $38.6244 (rounded to two decimal places)
Tip ≈ $38.62
Now, add the food bill and the tip to find the total cost before sales tax:
Total cost = Food bill + Tip
Total cost = $214.58 + $38.62
Total cost = $253.20
Therefore, the cost of the food bill and tip before sales tax was approximately $253.20.
Find the x- and y-intercepts for the linear equation.
7x + 9y = 63
Answer:
x-intercept: (9,0)
y-intercept: (0,7)
Step-by-step explanation:
x-intercept: when y = 0
7x + 9(0) = 63
7x = 63
x = 9
y-intercept: when x = 0
7(0) + 9y = 63
y = 7
Answer:
x-intercept : (9,0)
y-intercept : (0,7)
Step-by-step explanation: To find the x-intercept, substitute in 0 for y and solve for x. To find the y-intercept, substitute in 0 for x and solve for y.
Hope this helps you out! ☺
Only question 10. please
Nola was selling tickets. At the end of the evening she picked up the cash box and noticed a dollar on the floor and wonder if it was supposed to be in the box
The price of tickets for the Dance was 1$(for individuals) or 2 tickets for 8$(for couples) she looked inside the cash box and found 200$ and ticket subs for the 47 students in attendance, does the dollar nola found on the ground belong in the box ?
Let X = Individual ticket and y = couples tickets.
Set up two equations:
X + Y = 47 tickets sold
1X + 8y = $200
Rewrite the first equation as X = 47-y
Replace x in the second equation and solve:
1(47-y) = 8y = 200
Simplify:
47 - y + 8y = 200
Combine like terms:
47 +7y = 200
Subtract 47 from both sides:
7y = 153
Divide both sides by 7:
y = 153 / 7
y = 21.857
Because Y does not equal a whole number, the extra dollar does belong in the box.
Final answer:
Given the ticket prices and attendance, total sales amount to $71. The extra dollar found doesn't fit this, so it shouldn't be in the box.
Explanation:
To determine if the dollar Nola found belongs in the box, let's analyze the ticket sales. With individual tickets priced at $1 and couples tickets at $8 for two, we can calculate the total number of tickets sold. If there were 47 students in attendance, and assuming each couple bought the couples' ticket deal, the number of individual tickets sold would be (47 - number of couples) * 2 + number of couples.
Since each couple buys two tickets, the number of couples is 47 / 2 = 23.5, which means there were 23 couples and 1 individual. So, the total number of tickets sold is (47 - 23) * 2 + 23 = 71. Thus, the revenue from ticket sales should be $71.
If the cash box contains $200 and ticket sales total $71, then the extra dollar Nola found does not belong in the box. It's possible someone overpaid for a ticket or dropped a dollar accidentally, but it shouldn't be included in the revenue from ticket sales. Therefore, Nola should keep the dollar aside, perhaps to return it to its rightful owner.
Which phrase does NOT have the same meaning as -3/5?
A. Negative 3 times a number w
B. Negative 3 divided by a number w
C. A number w divided into negative 3
D. The quotient of negative 3 and a number w
Answer:
A
Step-by-step explanation:
You are dividing not multiplying
Answer the question Pls
Answer:
The first option:
Prove that the area of the two small squares combined is equal to the area of the larger square
Step-by-step explanation:
The Pythagorean Theorem in an equation would be:
a² + b² = c²
Where:
c = hypotenuse (longest side of the triangle)
a and b = two other legs of the triangle
The theorem states that the sum of the squares of the shorter sides is equal to the square of the longest side in a right triangle or a triangle that has a right angle (90°)
Let's say that we have a right triangle with the lengths of 3cm , 4cm, and 5cm.
(3cm)² + (4cm)² = (5cm)²
9cm² + 16cm² = 25cm²
25cm² = 25cm²
Look at the attached image to see what happens when we draw the squares of each leg.
Which best explains why -3 2/3 is a rational number
Step-by-step explanation:
it can be expressed in the form of p/q where p and q are co primes,
the basic definition of rational number
Answer:
it can be expressed as a fraction
Step-by-step explanation:
A town in east Texas received 20 inches of rain in two weeks. If it kept raining at this rate for a 31−day month, how much rain did the town receive? Round your answer to the nearest inch. The town received approximately inches of rain during a 31−day month
Answer:
The town received approximately 44.29 inches of rain during a 31−day month.
Step-by-step explanation:
Given;
The amount of rain for two weeks = 20 inches.
For finding amount of rain for 31 days, we use the unitary method.
∵ Amount of rain for 14 days = [tex]20\ inches[/tex]
∴ Amount of rain for 1 day = [tex]\frac{20\ inches}{14}[/tex]
∴ Amount of rain for 31 day = [tex]\frac{20\ inches}{14}\times31=\frac{620}{14}\ in=44.285\approx 44.29\ in[/tex]
Hence the amount of rain the town received is 44.29 inches approximately.