A family has a unique pattern in their tile flooring on the patio. An image of one of the tiles is shown.
A quadrilateral with a line segment drawn from the bottom vertex and perpendicular to the top that is 5 centimeters. The right vertical side is labeled 3 centimeters. The portion of the top from the left vertex to the perpendicular segment is 5 centimeters. There is a horizontal segment from the left side that intersects the perpendicular vertical line segment and is labeled 6 centimeters.
What is the area of the tile shown?
53 cm2
45.5 cm2
42.5 cm2
36.5 cm2
Answers
Answer:
The area of the tile can be found by dividing it into two triangles and one trapezoid. The area of the trapezoid is (5+3)/2 x 6 = 24 cm^2. The area of the two triangles can be found using the height of 5 cm and the base of 6 cm and 2 cm, respectively. The total area is then 24 + (5 x 6)/2 + (5 x 2)/2 = 24 + 15 + 5 = 44 cm^2. Therefore, the closest answer choice is 42.5 cm^2. The answer is (C) 42.5 cm^2.
Step-by-step explanation:
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Related Questions
Suppose you invested $17,400 into an account at a rate of 2. 5% compounded annually. Write the compound interest formula that models this investment scenario, then find the balance of the investment after 8 years
Answers
The compound interest formula that models this investment scenario is A = \(P(1 + r/n)^{(nt)\) ,the balance of the investment after 8 years is $20,419.05.
The compound interest formula can be used to calculate the balance of an investment over time when the interest is compounded annually. The formula is:
A = \(P(1 + r/n)^{(nt)\)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
For this problem, we have P = $17,400, r = 2.5% = 0.025, n = 1 (since the interest is compounded annually), and t = 8 years. Plugging these values into the formula, we get:
A = $17,400(1 + 0.025/1)⁸
A = $17,400(1.025)⁸
A = $20,419.05
This means that the investment has earned $20,419.05 - $17,400 = $3,019.05 in compound interest over 8 years. This shows the power of compounding interest, as the interest earned each year is added to the principal and earns additional interest in subsequent years.
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The price of a watch was increased by 20% to £162. What was the price before the increase?
Answers
Answer:
= £135
Step-by-step explanation:
Original price = 100%
Percentage increase = 20%
New price = 100% + 20% = 120%
If 120% = £162
What about 100% = ?
= (100 x 162) ÷ 120
= 16200 ÷ 120
= £135
Answer:
sh.129.60
Step-by-step explanation:
20%...€162
€162 times 80 over 100
€81 times 8....over 5
€648 divide 5
......ans...... €129.60
the ratio of zoes money to yolansa's is 3:7. Yolanda has $64 more than Zoe. If Yolanda gives (1)/(4) to zoe what is the new ratio of zoes money to yolanda
Answers
New Ratio = 19:21
This deals with ratios, fractions and basic arithmetics'.
The ratio of money of zoe to yolansa's own is 3:7.Thus, the total part for each of them will be;
Zoe; 3/10
Yolansa; 7/10
If the total money they both have is x, then they will have;Zoe; 3x/10
Yolansa; 7x/10
Now,Yolanda has $64 more than Zoe. Thus;\(\frac{7x}{10}\) - \(\frac{3x}{10}\) = 64
Simplifying this gives;
\(\frac{4x}{10}\) = 64
Rearranging gives;
4x = 640
x = 640/4
x = $160
Thus, originally;
Yolanda has \(\frac{7}{10}\) × 160 = $112
Zoe has \(\frac{3}{10}\) × 160 = $48
Yolanda now gives \(\frac{1}{4}\) of her money to Zoe.
Thus, she gives out \(\frac{1}{4}\) × 112 = $28
Thus,
Zoe now has; 48 + 28 = $76
Yolanda now has; 112 - 28 = $84
New ratio of zoes money to yolandas money is;76:84
Simplifying this ratio gives;
19:21
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We are required to find the new ratio of zoes money to yolanda
the new ratio of zoes money to yolanda is 19:21
Ratio of zoes money to Yolanda's = 3:7.
Zoes = 3
Yolanda = 7
Total ratio = 3 + 7 = 10
let
x = Total money
Zoes = 3/10
Thus, the total part for each of them will be;
Zoe= 3/10x
Yolanda = 7/10x
Yolanda has $64 more than Zoe.
Therefore,
7/10x - 3/10x = 64
(7x - 3x) / 10 = 64
4x/10 = 64
cross product
4x = 64 × 10
4x = 640
divide both sides by 4
x = 640/4
x = $160
Original share:
Zoe = 3/10x
= 3/10 × 160
= 0.3 × 160
= $48
Yolanda = 7/10x
= 7/10 × 160
= 0.7 × 160
= $112
If Yolanda gives 1/4 to zoe
= 1/4 of 112
= 1/4 × 112
= 0.25 × 112
= $28
New Zoe's share = $48 + $28
= $76
New Yolanda's share = $112 - $28
= $84
Their new ratio = 76 : 84
divide by 4
= 19 : 21
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whats 2/3x+3/7=1-4/7x
Answers
Answer:
\(x=6/13\)
Step-by-step explanation:
So we have:
\(\frac{2}{3}x+\frac{3}{7}=1-\frac{4}{7}x\)
Remove all the fractions to make things simpler. To do so, we can multiply by the LCM of the denominators.
The denominators are 3 and 7.
The LCM of 3 and 7 is 21. Thus, multiply everything by 21:
\(21(\frac{2}{3}x+\frac{3}{7})=21(1-\frac{4}{7}x)\)
Distribute:
\(\frac{42}{3}x+\frac{63}{7}=21-\frac{84}{7}x\)
Simplify the fractions:
\(14x+9=21-12x\)
Add 12x to both sides:
\(26x+9=21\)
Subtract 9 from both sides:
\(26x=12\)
Divide both sides by 26:
\(x=12/26\)
Reduce:
\(x=6/13\)
And that's our answer :)
Answer: x = 6/13
find the common denominator (21);
14/21x + 9/21 = 21/21 - 12/21x
move your variable to one side and your constants to the other:
26/21x = 12/21
multiply all by 21/26 to get a lone x:
x = 12/26
simplify:
x = 6/13
Combine like terms for the following expression:
6x+2x−8−3
Answers
Answer:
8x-11
Step-by-step explanation:
Answer:
8x-11
Step-by-step explanation:
6x and 2x combine to make 8x and -8 and -3 combine to make -11.
3) Kites are manufactured by identical firms. Each firm's total and marginal cost functions for weekly production are given by TC=0.01q2+100MC=0.02q In the long-run equilibrium, how many kites will each firm produce? Describe the long-run supply curve for kites. a. Suppose the weekly demand for kites is given by QD=4000−1000P. How many kites will be sold? How many firms will there be in the kite industry? b. Suppose that the weekly demand for kites suddenly goes up to QD=5000−500P. In the very short run, when it is impossible to manufacture any more kites than those produced for that weck, what will the price of kiter be? How much profit will cach kite maker cam? In the market short run, what will the price of kites be? How much profit will cach kite maker carn'? d. In the market long run, what will the price of kites be? How many new firms will enter the kite-making industry? How much profit will they carr? Be sure to show your work.
Answers
To determine the long-run equilibrium for each firm's production of kites, we need to find the level of output where marginal cost (MC) equals marginal revenue (MR). In perfect competition, firms maximize their profit by producing at the level where MC = MR.
Given the marginal cost function MC = 0.02q, we equate it to the marginal revenue, which is equal to the market price (P) in perfect competition. So, we have:
0.02q = P
Now, we need to find the price at which the quantity demanded (QD) equals the quantity supplied (QS) in the market. To do this, we set QD equal to QS:
QD = QS
Substituting the demand function QD = 4000 - 1000P into the equation, we get:
4000 - 1000P = QS
Now, we can solve for the price (P):
4000 - 1000P = 0.02q
Simplifying further:
1000P = 4000 - 0.02q
P = (4000 - 0.02q) / 1000
Now, we can substitute the value of P into the equation for MC to find the level of output (q):
0.02q = P
0.02q = (4000 - 0.02q) / 1000
Solving for q:
0.02q = 4 - 0.00002q
1.00002q = 4
q = 4 / 1.00002
q ≈ 3999.8
In the long-run equilibrium, each firm will produce approximately 3999.8 kites.
The long-run supply curve for kites in perfect competition is horizontal at the minimum average total cost (ATC) of the firms. This is because in the long run, firms have enough time to adjust their inputs and optimize their production processes, leading to production at the lowest possible cost.
a. To find the number of kites sold and the number of firms in the kite industry, we substitute the price (P) into the demand function QD = 4000 - 1000P:
QD = 4000 - 1000P
QD = 4000 - 1000(4000 - 0.02q) / 1000
Simplifying:
QD = 4000 - 4000 + 0.02q
QD = 0.02q
Since q represents the output per firm and we know that each firm produces around 3999.8 kites, we can substitute this value:
QD = 0.02 * 3999.8
QD ≈ 79.996
Approximately 80 kites will be sold. To find the number of firms in the kite industry, we divide the total quantity supplied by the output per firm:
QS = 3999.8
Number of firms = QS / q
Number of firms = 3999.8 / 3999.8
Number of firms = 1
b. In the very short run, when it is impossible to manufacture any more kites than those produced for that week, the price of kites will be determined by the demand and supply conditions. With the demand function QD = 5000 - 500P, we can find the equilibrium price:
QD = QS
5000 - 500P = QS
Substituting the supply quantity from above:
5000 - 500P = 3999.8
Solving for P:
500P = 5000 - 3999.8
Last month of the visitors at the zoo saw the panda. This month of the visitors at the zoo saw the panda. What fraction more of the visitors saw the panda this month? Simplify the answer if possible.
A. 1/12
B. 1/4
C. 1/6
D. 3/4
Answers
Answer:
There is no way of finding out.
Step-by-step explanation:
3/4(12a+8) =27.6
3/4 Multiplied by () + (multiply by 8) = ()
A =?
PLEASE HELP THIS IS MY LAST QUESTION!!!!!
Answers
The value of a in the equation 3/4(12a+8) =27.6 is
2.4
What is linear equation?A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.
The equation 3/4(12a+8) =27.6 can be solved as follow:
3/4(12a+8) =27.6
multiply both sides by 4/3
12a + 8 = 27.6 × 4/3
12a + 8 = 36.8
collect like terms
12a = 36.8 - 8
12a = 28.8
divide both sides by 12
a = 28.8/12
a = 2.4
therefore the value of a is 2.4
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A truck that is 12 ft. by 10 ft. by 15 ft. carries cube
shaped boxes that have a length of 2 ft. How many
of these boxes can this truck hold?
Answers
Answer: 284 boxes in the shape of the cubes can be fitted in the truck.
Step-by-step explanation:
If a truck is fitted fully with the cubes, volume of the truck will be equal to the volume of cubes to be fitted.
Dimensions of the truck given in the question,
Length = 10 feet, width = 12 feet and height = 8 feet
Since, volume of a cuboid (shape of a truck) is given by the expression,
Volume = Length × Width × Height
Therefore, volume of the truck = 10 × 12 × 8
= 960 feet³
Volume of a cube is given by the expression,
Volume = (side)³
Therefore, volume of the cube with side length 1.5 feet will be,
Volume of a cube = (1.5)³
= 3.375 cubic feet
Let the number of cubes fitted in the truck = x
Therefore, volume of 'x' cubes = (3.375x) feet³
Since, volume of the truck = Volume of the cubes fitted
3.375x = 960
x = 960/3.375
x = 284.44
x ≈ 284
Therefore, 284 boxes in the shape of the cubes can be fitted in the truck.
Answer:
225 boxes
Step-by-step explanation:
1. Calculate the volume of the container:
\(v_{c} =(12)(10)(15)=1800ft^{3}\)
2. Calculate the volume of the boxes:
\(v_{b} =(2)^{3} =8ft^{3}\)
3. Calculate the number of boxes that fit in the container:
\(boxes=\frac{1800}{8} =225\)
Hope this helps
Mr. rosario is preparing a 12.4 pound veal roast for 16 guests. if each serving is about the same, how much veal roast will each guest receive?
Answers
Each guest receive 352 gm veal roast.
What is a pound used to measure?The pound is a unit of measurement used in the U. S. customary system and the British imperial system to measure weight. One familiar use of pounds is measuring how much a person weights.
We know that,
1 pound = 453.592 gm
12.4 pound = ? gm
12.4 pound = 12.4×453.592
= 5624.541 gm
Total number of guest = 16
Then,
Each guest receive veal roast = \(\frac{5624.541 gm}{16}\)
= 352 gm
Hence, Each guest receive veal roast is 352 gm.
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Use the distributive property to rewrite this expression.
4(c+8)
Answers
Answer: 4c+32
Step-by-step explanation:
use cylindrical coordinates. evaluate x2 dv, e where e is the solid that lies within the cylinder x2 y2 = 4, above the plane z = 0, and below the cone z2 = 36x2 36y2.
Answers
Using cylindrical coordinates ∫∫∫ (r^3cos^2θ) dz dr dθ, where r ranges from 0 to 2, θ ranges from 0 to 2π, and z ranges from 0 to √(36r^2).
To evaluate the integral ∫∫∫ x^2 dV over the solid e, using cylindrical coordinates, we need to express the integral in terms of cylindrical coordinates and determine the appropriate bounds for the variables.
In cylindrical coordinates, the solid e can be defined as follows:
Radius: r ranges from 0 to 2 (from x^2 + y^2 = 4, taking the square root).
Angle: θ ranges from 0 to 2π (full revolution around the z-axis).
Height: z ranges from 0 to the height of the cone, which is determined by z^2 = 36x^2 + 36y^2.
To convert the integral, we need to express x^2 in terms of cylindrical coordinates:
x^2 = (rcosθ)^2 = r^2cos^2θ
The integral in cylindrical coordinates becomes:
∫∫∫ (r^2cos^2θ) r dz dr dθ
Now we can determine the bounds for the variables:
r ranges from 0 to 2.
θ ranges from 0 to 2π.
z ranges from 0 to the height of the cone, which can be determined by setting z^2 = 36r^2.
Substituting the bounds and integrating, we can evaluate the integral to find the desired result.
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The base of the pyramid is a regular hexagon. Find the volume of the pyramid. Round your answer to the nearest tenth.
pls pls help answer
Answers
Answer:
216.5
Step-by-step explanation:
Pyramid volume=(area of the base)*(height)/3
Pyramid volume=(area of hexagon with side 5)*(height)/3
Area of hexagon is 3*sqrt(3)*(side)^2/2=75*sqrt(3)/2
Pyramid volume=10*(75*sqrt(3))/(2*3)=216.5
2 + 14 + 2 − 12 = -4
Answers
express 4.684684684... as a rational number, in the form pq where p and q have no common factors.
Answers
4.684684684... can be expressed as a rational number 4684/999, where 4684 (p) and 999 (q) have no common factors.
To express 4.684684684... as a rational number in the form p/q, where p and q have no common factors and 4.684684684... can be expressed as a rational number 4684/999, where 4684 (p) and 999 (q) have no common factors. follow these steps:
Subtract the integer part (4) from the number.
4.684684684... - 4 = 0.684684684...
Let x represent the decimal part.
x = 0.684684684...
Note the repeating pattern (684) and multiply x by 1000 to shift the pattern to the left.
1000x = 684.684684...
Subtract the original x equation from the new equation to eliminate the repeating decimals.
1000x - x = 684.684684... - 0.684684684...
999x = 684
Solve for x by dividing both sides by 999.
x = 684/999
Since there are no common factors between 684 and 999, this is the simplified rational number. Now add back the integer part (4) to get the final answer.
4 + 684/999 = 4684/999
So, 4.684684684... can be expressed as a rational number 4684/999, where 4684 (p) and 999 (q) have no common factors.
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SOLVE FOR Y
2x-y=3
pls help
Answers
Answer:
x = 3+y/2
Step-by-step explanation:
Ralph’s current annual income is $32,000. Using the graph shown below, Ralph has calculated that he will have to pay a total of $9,280 in federal taxes this year. Which of the following statements is true?
A circle graph titled Simpson's Annual Expenses. Federal taxes, 24 percent; State/local taxes, 10 percent; Housing and household, 23 percent; Food, 6 percent; Medical care, 8 percent; Transportation, 11 percent; Recreation, 3 percent; Clothing, 5 percent; Other, 10 percent.
a.
Ralph’s calculation is too high, he has only calculated state taxes.
b.
Ralph’s calculation is too high, his calculation includes state taxes.
c.
Ralph’s calculation is correct, he will have to pay $9,280 in federal taxes this year.
d.
Ralph’s calculation is too low, he did not include medical care in his calculation.What is gross pay
Answers
c.
Step-by-step explanation:
Ralph’s calculation is correct, he will have to pay $9,280 in federal taxes this year.
Ralph ’s calculation is too high, as his calculation includes state taxes.
CalculusGiven that Ralph's current annual income is $32,000, and he has calculated that he will have to pay a total of $9,280 in federal taxes this year, knowing that he has to pay 24% in federal taxes, to determine if his calculation is correct you must perform the following calculation:
32000 x 0.24 = X7680=XTherefore, Ralph ’s calculation is too high, as his calculation includes state taxes.
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Please help, I’ll give brainliest as soon as I can! I need an answer quick
Answers
Step-by-step explanation: 0-9-8+xyz
True or False: Arrhenius equation can rearrange to give a linear relationship by taking the natural logarithm.
Answers
The statement " Arrhenius equation can rearrange to give a linear relationship by taking the natural logarithm." is true.
The Arrhenius equation is an important tool in chemical kinetics that describes the temperature dependence of reaction rates. It relates the rate constant of a chemical reaction to the activation energy and temperature at which the reaction occurs. The equation is given by:
\($k = A\mathrm{e}^{-\frac{E_a}{RT}}$\)
where \($k$\) is the rate constant,\($A$\)is the pre-exponential factor, \($E_a$\) is the activation energy, \($R$\)is the gas constant, and \($T$\) is the temperature in Kelvin.
Although this equation is useful, it is not always easy to interpret experimentally. The natural logarithm of both sides of the equation can be taken, resulting in the following equation:
\($\ln k = \ln A - \frac{E_a}{RT}$\)
This equation can be rearranged into the form of a linear equation,
\($y = mx + b$\),
by defining:
\($y = \ln k$\)
\($m = -\frac{E_a}{R}$\)
\($x = \frac{1}{T}$\)
\($b = \ln A$\)
Therefore, we have:
\($y = mx + b$\)
which can be plotted as a straight line. By analyzing the slope and intercept of this line, we can determine the values of the activation energy and pre-exponential factor, which are important parameters in understanding the kinetics of a chemical reaction.
In summary, the Arrhenius equation can be rearranged to give a linear relationship by taking the natural logarithm of both sides of the equation. This linear form is often useful for analyzing experimental data and determining the activation energy and pre-exponential factor of a reaction, which are important parameters in understanding the kinetics of chemical reactions.
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how do I find the length of the side of a pentagon?
Answers
Answer:
The equation to find the length of one side of a pentagon using the radius looks like this: {display style side length=2rsin ({frac {180} {n}})}. It may look a little complicated, but you can easily plug in the numbers that you already know to simplify the equation and find the length of a side. r represents the radius of the pentagon.
Solve each quadratic equation by factoring.
Show steps pls
8x^2 - 4x = 2x +9
Answers
Answer:
hope these help........
Answer:
\(x = \frac{-3}{4}\) or \(x = \frac{3}{2}\)
Step-by-step explanation:
\(8x^{2} - 4x = 2x +9\)
Okay, let's put everything on one side.
\(8x^{2} -4x-2x-9=0\)
As you can see we can subtract -4x from -2x now leaving us with -6x.
\(8x^{2} -6x-9=0\)
Now there are multiple ways to solve a polynomial, but we can go with a way to factor them the easy way. We need to multiply the leading coefficient with the ending term because there is no common number we can factor out of these guys.
\(8(-9)=-72\)
We multiplied 8 and -9 together which gave us -72. With this, we need to find two numbers that add to the middle term, -6, but will multiply to -72. This is going to be 6 and -12. Now we are going to replace the middle coefficient with those two numbers.
\(8x^{2} + 6x - 12x- 9= 0\)
(Keep in mind the equation has not changed)
Now we are going to do something called factoring by grouping, with the first two terms, 8x^2 and 6x we take out the GCF (greatest common factor):
\(2x(4x+3)\)
so let's do the same between the last two.
\(-3(4x+3)\)
So when putting them together we have,
\(2x(4x+3)-3(4x+3)=0\)
Now let's factor here, (4x+3) can be taken out and 2x as well as -3 can be put together.
\((4x+3) (2x-3)=0\)
Then we set these factors to zero to get our x answer:
\(4x+3=0\)
\(2x-3=0\)
And we solve, which leaves us with:
\(x = \frac{-3}{4}\) or \(x = \frac{3}{2}\)
A person invests 8000 dollars in a bank. The bank pays 6.25% interest compounded daily. To the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 10800 dollars
Answers
Answer:
4.8 years
Step-by-step explanation:
10 800 = 8000 e^(i t) i = decimal interest per year t = years
10800/8000 = e^(.0625 t) ln both sides
.300 = .0625 t
t = 4.8 years
Given the figure below find the values of x and z
(12x +2)°
94°
z°
Answers
Answer:
z = 94
x = 7
Step-by-step explanation:
The angle with a measure of 94° and the angle labeled "z" are vertical angles
Vertical angles are congruent ( equal to each other )
Which means that because "z"s vertical angle = 94° z also equals 94°
The angle labeled with the expression ( 12x + 2 ) and the angle with a measure of 94° are supplementary angles
Supplementary angles add up to equal 180
Hence, 94 + 12x + 2 = 180
^ (Note that this is the equation we will use to solve for x )
Now we solve for x
94 + 12x + 2 = 180
step 1 combine like terms
94 + 2 = 96
we now have 180 = 96 + 12x
step 2 subtract 96 from each side
180 - 96 = 84
96 - 96 cancels out
we now have 84 = 12x
step 3 divide each side by 12
84 / 12 = 7
12x / 12 = x
we're left with x = 12
Answer:
z = 94
x = 7
Step-by-step explanation:
I have no Idea of what to write, my teacher never taught me this. Please help
Answers
Answer:
24
Step-by-step explanation:
hope this help if it dont sorry i tried
The following excerpt comes from the international bottled water association:
"in 2012, total u.s. bottled water consumption increased to 9.67 billion gallons, up from 9.1 billion gallons in 2011. in fact, 2012's consumption growth was the strongest it has been in five years. in addition, per-capita consumption is up 5.3 percent in 2012, with every person in america drinking an average of 30.8 gallons of bottled water last year. bottled water increased in absolute volume more than any other beverage category in the u.s. bottled water sales increased by 6.7 percent in 2012, and now total $11.8 billion."
(i) quantify the bottled water consumption increase from 2011 to 2012 in terms of relative change. (express your answer as a percent rounded correctly to the nearest tenth of a percent.)
%
(ii) if bottled water sales are up by 6.7% in 2012 to a total of $11.8 billion, what were the sales in 2011? (express your answer in billions of dollars rounded correctly to the nearest tenth.)
$
11.1
billion
(iii) estimate the per capita consumption from 2011 from data in the statement. (express your answer rounded correctly to the nearest tenth of a gallon.)
gallons
Answers
Using proportions, the estimate of the per capita consumption from 2011 was 29.25 gallons.
What is a proportion?A proportion is a fraction of the total amount, the measures are related using a rule of three.
The amount in 2012 was an increase of 5.3% from 2011,
105.3% = 1.053 of x,
The consumption per capita was 30.8 gallons,
hence:
1.053x = 30.8
x = 30.8/1.053
x = 29.25 gallons.
(ii) The sales in 2011
= 9.1 billion gallons in 2011.
(iii) The per capita consumption from 2011
= 9.1 / 11.8
= 0.77 billion
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What is the surface area?
in
- 5 in
Answers
Answer:
94 ft
Step-by-step explanation:
Is 3x + 2y = 6 a function??
A. No, the equation is not a function; there is an x-value that produces two distinct y-values.
B. Yes, the equation is a function; the graph of the line passes the vertical line test.
C. Yes, the equation represents a function because there is both an x and a y.
Answers
Answer:
B :)
Step-by-step explanation:
Olivia is jogging at a speed of 1.6 meters per second. What is her speed in kilometers per hour?
Answers
Answer:
.6 se
Step-by-step explanation:
Answer:
5.76
Step-by-step explanation:
a p e x
f 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? 5 if 6 bottles are randomly selected, what is the probability that all of them are the same variety?
Answers
The probability of selecting all bottles of the same variety is P(A) = 6/46656 = 0.00013.
The probability of selecting two bottles of each variety when randomly selecting 6 bottles can be calculated using the formula P(A) = n(A) / n(S), where n(A) is the number of possible ways to select two bottles of each variety and n(S) is the total number of possible ways to select 6 bottles. In this case, n(A) is equal to 6! / (2!2!2!) = 90 and n(S) is equal to 6^6 = 46656. So, the probability of selecting two bottles of each variety is P(A) = 90/46656 = 0.0019.
The probability of selecting all bottles of the same variety when randomly selecting 6 bottles can be calculated using the same formula. In this case, n(A) is equal to 6, which is the number of ways to select 6 bottles of the same variety, and n(S) is equal to 46656. So, the probability of selecting all bottles of the same variety is P(A) = 6/46656 = 0.00013.
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Determine a definite integral that represents the arc length of r = 4 cos(theta), 0 ≤ 0 ≤ π/2 Be sure to show steps of work on your paper for setting up and simplifying the integral. You do NOT need to evaluate the integral.
Answers
The definite integral that represents the arc length of the polar curve r = 4cos(theta) over the interval 0 ≤ theta ≤ π/2 is ∫(0 to π/2) 4 dθ.
To determine the definite integral that represents the arc length of the polar curve r = 4cos(theta) over the interval 0 ≤ theta ≤ π/2, we can use the arc length formula for polar curves:
L = ∫(a to b) √(r^2 + (dr/dθ)^2) dθ
In this case, the interval is from 0 to π/2.
Let's start by finding the derivative dr/dθ:
dr/dθ = -4sin(theta)
Now, substitute the values into the arc length formula:
L = ∫(0 to π/2) √(r^2 + (dr/dθ)^2) dθ
= ∫(0 to π/2) √(4cos^2(theta) + (-4sin(theta))^2) dθ
= ∫(0 to π/2) √(16cos^2(theta) + 16sin^2(theta)) dθ
= ∫(0 to π/2) √(16(cos^2(theta) + sin^2(theta))) dθ
= ∫(0 to π/2) √(16) dθ
= ∫(0 to π/2) 4 dθ
Simplifying the integral, we have:
L = 4 ∫(0 to π/2) dθ
Therefore, the definite integral that represents the arc length of the polar curve r = 4cos(theta) over the interval 0 ≤ theta ≤ π/2 is ∫(0 to π/2) 4 dθ.
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