Answer: About 20.82 liters.
Step-by-step explanation:
To go from gallons to liters, we will multiply by 3.785. Since we are in half gallons, we will multiply 11 by 0.5 for 5.5 full gallons. Now, we will multiply;
5.5 gallons * 3.785 = 20.8198 ≈ 20.82 liters
Please help, I don't quite understand.
Answer:
[tex]y=-\frac{3}{4}x-2[/tex]
Step-by-step explanation:
Substitute into point-slope form and rearrange to slope-intercept form.
[tex]y+8=-\frac{3}{4}(x-8) \\ \\ y+8=-\frac{3}{4}x+6 \\ \\ y=-\frac{3}{4}x-2[/tex]
solve
pleaseeeeeeeeeeee
The equation can be written as - 8x + 12 = 8x + 12 which means it has infinite solutions. So, there are infinite values of [x] for which parameters are equal.
What is Parallelogram? What is triangle?In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. A triangle is a polygon with three edges and three vertices. The sum of all the angles of a triangle is 180 degrees. Mathematically -
∠x + ∠y + ∠z = 180°
There are different types of triangles such as -
equilateral triangle , scalene triangle , isosceles triangle etc.
Given is a square and a triangle as shown.
For equal perimeter -
4(2x + 3) = (2x - 2) + 2(3x + 7)
8x + 12 = 2x - 2 + 6x + 14
8x + 12 = 8x + 12
The equation can be written as -
8x + 12 = 8x + 12
which means it has infinite solutions.
So, there are infinite values of [x] for which parameters are equal.
Therefore, the equation can be written as - 8x + 12 = 8x + 12 which means it has infinite solutions. So, there are infinite values of [x] for which parameters are equal.
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HELP!!! a wooden block in a set of building blocks is a rectangular prism with a volume of 168 cubic centimeters. the block is x2 centimeters tall, (x-3) centimeters long, and (x-4) centimeters wide. find the dimensions of the block.
The dimensions of the wooden block are 14cm tall, 4cm long and 3 cm wide.
What is prism?In three dimensional space prism is a polyhedron where two ends are similar.
Given that volume of rectangular prism is 168 cubic centimeters.
And the wooden block is 2x cm tall, (x-4) cm wide and (x-3) cm long.
And also given that a wooden block in a set of building blocks is a rectangular prism.
That means,
Volume of wooden block = Volume of rectangular prism = 168 cm³
Volume of wooden block ;
length × width × height = 168 cm³
Substituting given values in above expression,
2x (x-3) (x-4) = 168 cm³
2x (x²-7x + 12) = 168 cm³
2x³ - 14x²+ 24x = 168 cm³
Simplifying the above equation;
2x³ - 14x²+ 24x- 168 = 0
x³ - 7x²+ 12x- 84 = 0
The factor of above cubic equation is x = 7.
Now the 2x = 14 , x-3 = 4, x-4 = 3.
Therefore the dimensions of the wooden block are 14cm tall, 4cm long and 3 cm wide.
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If a coil of cable ,50m in length has a conductor resistance of 0. 275W,calculate the value for 800m of the same cable
If a coil of cable ,50m in length has a conductor resistance of 0. 275W,then the value of resistance for 800m of the same cable will be 4.4 ohms.
What is resistance?The opposition to current flow in an electrical circuit is measured by resistance. The Greek letter omega represents the unit of measurement for resistance, known as ohms. Georg Simon Ohm (1784–1854), a German physicist who investigated the connection between voltage, current, and resistance, is the name given to the unit of resistance.
what is conductance?the ratio of a conductor's current to potential difference, which measures how well a thing conducts electricity. This is the resistance' inverse, and it is measured in siemens.
as we are given with;
Length=50m
Resistance=0.275W
we got Formula;
Resistance=pL/A
p/A=R/A
p/A=0.275/50
=0.0055
Now for 800m wire
R= p/A(800)
R=0.0055x800
R=4.4 ohms
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On a given day, the proportion of workers from Company B who purchase a coffee from the company cafeteria is 0. 62 and the proportion of workers from Company C who purchase a coffee from the company cafeteria is 0. 71. A random sample of 40 workers was selected from Company B and another random sample of 40 workers was selected from Company C. The proportion of workers from Company B who purchased coffee was
The correct unit of measure for the mean of the sampling distribution of (pˆB - pˆC) is Workers , the correct option is (d) .
In the question ,
it is given that ,
the proportion of workers who purchase coffee from company B is = 0.62
the proportion of workers who purchase coffee from company C is = 0.71
the random sample taken from company B is = 40 workers
the random sample taken from company C is = 40 workers
we know that , a proportion is written as two equivalent fractions.
So , Each ratio compares the same units, percentage of workers purchasing the coffee in the restaurant,
hence, the unit of measure for the mean of sampling distribution is workers .
Therefore , the unit for the measurement for the mean is (d) Workers .
The given question is incomplete , the complete question is
On a given day, the proportion of workers from Company B who purchase a coffee from the company cafeteria is 0.62 and the proportion of workers from Company C who purchase a coffee from the company cafeteria is 0.71. A random sample of40 workers was selected from Company B and another random sample of 40 workers was selected from Company C. The proportion of workers from Company B who purchased coffee was pˆB=0.70 and the proportion of workers from Company C who purchased coffee was pˆC=0.75.
What is the correct unit of measure for the mean of the sampling distribution of pˆB - pˆC ?
(a) Days
(b) Dollars
(c) Companies
(d) Workers .
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can someone help me out here,im a little confused.
Three vertices of a square are A(1,2), B(4,2), and C(4,−1).
Plot all four vertices of the square.
Using a graphing calculator to plot the vertices, the points plotted gave a triangle
Vertices of a SquareThis can be said to be the location of the endpoints in a given square. The vertices of a square must be equidistant from one another. This is a place in geometry where common endpoints of two or more rays or line segments meet. Thus, in a square, the vertex is the point where two sides meet. It is each corner of a square. Thus, each square has 4 vertices.
In this problem, we have to use a graphing calculator and plot each points
In doing so, we have A(1, 2), B(4, 2) and C(4, -1)
Plotting the points, it gives a triangle instead of a square;
The point A and B are located in the first quadrant and point C is located in the second quadrant.
Kindly find the attached graph below
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At the beginning of the year, Juan had $60 in savings and saved an additional $15
each week thereafter. Mila started the year with $40 and saved $20 every week. Let J
represent the amount of money Juan has saved t weeks after the beginning of the
year and let M represent the amount of money Mila has saved t weeks after the
beginning of the year. Graph each function and determine the number of weeks after
the beginning of the year until Juan and Mila have the same amount of money saved.
Answer: They both meet up at $160
Step-by-step explanation:
40 + 20= 60 + 20 = 80 + 20 = 100 + 20 = 120 + 20 = 140 + 20 = 160
60 + 15 = 75 + 15 = 90 + 15 = 115 + 15 = 130 + 15 = 145+ 15 = 160
80% of what number is 16
a. PART
b.TOTAL
c.PERCENT
d. WRITE AND SOLE THE PROPORTION
Answer:
:)
Step-by-step exp
16/.8 = 20
Answer:
20
Step-by-step explanation:
ask urself what 80/100 is to 16 call the number x find xWhile working a summer job, Evelyn earned $198.10 in 7 hours. Find the rate of change.
Answer:
$221
Step-by-step explanation:
An item is regularly priced at $83. It is on sale for 35% off the regular price.
Answer: $53.95
Step-by-step explanation:
Original Price: $83
Discount: 35%
35%=0.35
$83 x 35%, or 83 x 0.35
=29.05
$83-29.05
=$53.95
F(x) is defined by the equation f(x) = 5x² - 9x-12. What effect will multiplying
f(x) by 0.5 have on the graph?
A. The graph will be compressed vertically.
B. The graph will be stretched horizontally.
C. The graph will be stretched vertically.
D. The graph will be compressed horizontally.
F(x) is defined by the equation f(x) = 5x² - 9x-12, the effect will multiplying
f(x) by 0.5 have on the graph ,then the graph will be compressed vertically.
What is a compressed graph?A transformation in which all distances on the coordinate plane are shortened by multiplying either all x-coordinates (horizontal compression) or all y-coordinates (vertical compression) of a graph by a common factor less than 1.
Given that;
f(x) = 5x² - 9x-12
Effect will multiplying f(x) by 0.5 , After the transformation
g(x) = (0.5) f(x)
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression.After transformation the graph will be composed vertically.
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PLEASE HURRY!!
Look at the picture of a scaffold used to support construction workers. The height of the scaffold can be changed by adjusting two slanting rods, one of which, labeled PR, is shown:
A support structure is shown in which a right triangle PQR is formed with the right angle at Q. The length of PQ is shown as 14 feet, and the length of QR is shown as 9 feet.
Part A: What is the approximate length of rod PR? Round your answer to the nearest hundredth. Explain how you found your answer stating the theorem you used. Show all your work. (5 points)
Part B: The length of rod PR is adjusted to 18 feet. If width PQ remains the same, what is the approximate new height QR of the scaffold? Round your answer to the nearest hundredth. Show all your work. (5 points)
Approximate new height QR of the scaffold would be 11.31 Fee
What is Pythagoras theorem?
The pythagorean theorem or Pythagoras' theorem may be a elementary relation in geometry between the 3 sides of a triangle.
Main Body:
A) Here, We'll use "Pythagoras Theorem" which tells:
a² + b² = c²
So, PR² = PQ² + QR²
PR² = 14² + 9²
PR² = 196 + 81
PR = √277
In short, approximate length of rod PR would be 16.64 Feet.
B) Again, Use the Pythagoras Theorem,
c² - a² = b²
18² - 14² = b²
b² = 324 - 196
b = √128
b = 11.31
Hence , approximate new height QR of the scaffold would be 11.31 Feet
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Let f(x)=x3 and g(x)=13x.
Use the composition of functions to determine if f(x) and g(x) are inverse functions.
Drag and drop the answers into the boxes to correctly complete each statement.
The functions f(g(x)) and g(f(x)) are not inverse functions
How to determine if the functions are inverse functions?From the question, we have the following parameters that can be used in our computation:
f(x) = x³
g(x) = 1/3x
Calculate the composite function f(g(x)).
So, we have
f(g(x)) = (1/3x)³
This gives
f(g(x)) = 1/27x³
Calculate the composite function g(f(x)).
So, we have
g(f(x)) = 1/3(x³)
This gives
g(f(x)) = 1/3x³
The functions f(g(x)) and g(f(x)) are not the same
Hence, they are not inverse functions
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15. Every year in Delaware there is a contest where people create cannons and catapults designed
to launch pumpkins as far in the air as possible. The equation y = 12 + 105x-16x² can be
used to represent the height, y, of a launched pumpkin, where x is the time in seconds that the
pumpkin has been in the air. What is the maximum height that the pumpkin reaches? How
many seconds have passed when the pumpkin hits the ground? (Hint: If the pumpkin hits the
ground, its height is 0 feet.)
(1 point)
To find the maximum height that the pumpkin reaches, we can set the derivative of the equation equal to 0 and solve for x. The derivative of y = 12 + 105x - 16x^2 is y' = 105 - 32x. Setting y' = 0, we get:
0 = 105 - 32x
32x = 105
x = 105/32
The maximum height is reached when x = 105/32 seconds. To find the maximum height, we can plug this value of x into the original equation:
y = 12 + 105(105/32) - 16(105/32)^2
= 12 + 1053.28125 - 163.28125^2
= 12 + 342.8125 - 106.25
= 248.5625
The maximum height that the pumpkin reaches is 248.5625 feet.
To find the number of seconds it takes for the pumpkin to hit the ground (when its height is 0), we can set y = 0 in the original equation and solve for x:
0 = 12 + 105x - 16x^2
16x^2 - 105x + 12 = 0
(4x - 3)(4x - 4) = 0
The solutions to this equation are x = 3/4 and x = 4/4. The second solution is extraneous, since it represents the time at which the pumpkin was launched (when x=0, y would be 12 feet, not 0 feet). Therefore, the number of seconds it takes for the pumpkin to hit the ground is x = 3/4 seconds.
This one is confusing please someone help:
1. give another name for plane p
2. name a line that intersects the plane
3. name two rays
4. name 3 points in plane p
1.You can find another name for plane P by using any three points in the
plane that are not on the same line. Hence, the another name for plane P is the plane GAC .
2. The line intersects the plane is EF
3. The two rays in the plane are CB , DB
4. A,B,C are three points in the plane p.
What is meant by plane?Spanned by two linearly independent vectors, a plane is a doubly ruled surface in two dimensions. Hyperplanes are the result of extending the plane to higher dimensions. The dihedral angle is the angle at which two planes intersect.
Using any three points on the plane that are not on the same line, you can come up with another name for plane P. Therefore, another name for plane P is GAC. The plane is intersected by the line at EF. CB and DB are the two rays in the plane. There are three points in the plane p, A, B, and C.
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HELP ASAP 100 POINTZZZZZZ
ollow the instructions below.
(a) Rewrite the percentage in the sentence below as a decimal. Decimal:
In a recent poll, 5% of the people surveyed were in favor of the new law.
(b) Rewrite the decimal in the sentence below as a percentage. Percent:%
The model was 0.274 of the size that the original was.
Decimal of 5% = 0.05, 0.274 to Percent:% 27.4%
What is percentage?A percentage is a figure or ratio that shows a portion of one hundred. A dimensionless connection between two numbers can be represented in a variety of ways, such as through ratios, fractions, and decimals. The sign "%" is frequently put after the number to indicate percentages.
What is decimal?A decimal is a number that is divided into two parts: a whole and a fraction. Between integers, decimal numbers are used to express the numerical value of complete and partially whole amounts.
a).
so we have to write this in decimal that is remove percentage
To remove % we have , divide it by 100
suppose x% to remove % we divide it by 100, x% = x/100
5% = 5/100 =0.05 which is in decimal
b). The model was 0.274 of the size that the original was. we need to add percentage so to add percentage multiple it by 100
making x to percentage
x*100 = x%
so 0.274 to % will be
0.274 *100
27.4%
Decimal of 5% = 0.05, 0.274 to Percent:% 27.4%
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If 80% of 120 students passed the test, and (1)/(4) of these students received an A on the test, how many students received an A?
A 20 students
B 22 students
C 24 students
D 26 students
24 students received an A on the test.
What is the system of linear equations?
A system of linear equations is a group of one or more linear equations that can be solved by using a simultaneous equation or elimination method. Therefore, by using the substitution method and writing the solution of the system in terms of coordinate point (x, y) = (-1, 1).
If 80% of 120 students passed the test, the number of students who passed is 120 * 80% = 96 students.
If (1)/(4) of these students received an A on the test, the number of students who received an A is (1)/(4) * 96 = 24 students.
Therefore, 24 students received an A on the test.
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statisticians have been able to show that the normal bell curve property also occurs when the sample size is 30 observations or fewer. true false
statisticians have been able to show that the normal bell curve property also occurs when the sample size is 30 observations or fewer: False.
Where in a normal curve can most observations be found?The majority of the observations are centred around the middle peak of the normal distribution, which is a continuous probability distribution that is symmetrical around its mean.
The normal distribution is a continuous probability distribution with symmetry around its mean, a central peak around which the majority of observations cluster, and a tapering off of probabilities in both directions for values that are farther from the mean.
When a result is significant, it simply implies that you may be certain that it is true, not that you merely got fortunate (or unlucky) in the sample selection.
Thus, the statement given in the question is false.
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domain:1,2,3,4,5 range:68,71,74,77,80 what is a reasonable domain for this
The reasonable domain is { 1,2,3,4,5 }
What are domain and range of a function ?
The range of values that we are permitted to enter into our function is known as the domain of a function. In a function like f, this set represents the x values (x). The collection of values that a function can take on is known as its range. The values that the function outputs when we enter an x value are in this set.
The collection of all feasible independent variable values is known as the mathematical domain. The set of all independent variable values that are conceivable yet make sense in the context at hand is known as the reasonable domain.
Here, the situation is not given.
So, the reasonable domain is { 1,2,3,4,5 }
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A teacher grades 27 tests in 3 hours.
find the constance of variation
Answer: 9
Step-by-step explanation:
27 divided by 3 is 9 so 9 is the constant variation
A customer pays an initial fee and a daily fee to rent a snowmobile. The total payment for 3 days is 92 dollars. The total payment for 5 days is 120 dollars. What is the daily fee?.
A snowmobile can be rented for $14 per day with a $50 initial fix cost.
What is equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
Here,
Let the initial fee be x and daily fee be y,
92=x+3y
120=x+5y
x+5y=120
x+3y=92
Subtract both,
2y=28
y=$14
x+5y=120
x+70=120
x=$50
The daily fee to rent a snowmobile is $14 per day with a initial fix fee of $50.
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You in-line skate from an elevation of 720 feet to an elevation of 750 feet in 30 minutes. Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour. Compare your rates of change in elevation.
Please help im stuck on this problem
Your friend's rate of change in elevation is 0.5 feet per minute more than your rate.
What is the formula for the rate of change in elevation?To find the height of the building (QR), we can use the angle of elevation formula tan θ=QR/PR.If you're looking for an exact number, use an altimeter, computer, or smartphone to calculate your current position above sea level. If you're trying to find a difference in elevation between 2 points of land, use tie a length of string between two posts and use a level to determine the difference.Let the area of the triangle be xy/2 and dA/dt becomes dA/dt= (xy' +yx')/2. To find y' use the Pythagorean theorem x2 +y2 =z2 where z is constant. So xx' +yy' =0 and y'=-xx'/y and substituting this into dA/dt gives dA/dt =(-x2x'/y +x'y)/2 = x'(y2 -x2)/2y.An angle of elevation is an angle with one horizontal arm, and one arm above horizontal. Usually an angle of elevation is less than or equal to 90°.Rate of change in elevation = elevation of distance/time
The elevation of 720 feet to an elevation of 750 feet in 30 minutes.
Your rate of change in elevation = 750 - 720 / 30
Your rate of change in elevation = 30 / 30
Your rate of change in elevation = 1 foot per minute
Your friend in-line skates from an elevation of 600 feet to an elevation of 690 feet in one hour.
Your friend's rate of change in elevation = 690 - 600 / 60
Your friend's rate of change in elevation = 90 / 60
Your friend's rate of change in elevation = 1.5 feet per minute
The difference in rate of change in elevation = 1.5 - 1 = 0.5 feet per minute
Thus, your friend's rate of change in elevation is 0.5 feet per minute more than your rate.
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Conider the algebraic expreion
32pq2−4p2q12pq−7
Find term that are not contant. A) 32p2q,−4p2q,12pq, 7
b) 32, -4, 12, 7
c) 32pq2,4pq2,12pq
d) 32pq2, −4p2q, 12pq
The correct part is (d) i.e. 32pq2, -4p2q and 12pq are not constants in the given algebraic expression.
What is a constant?
A constant is a value or number that is consistently the same, regardless of how it is expressed.
The algebraic expression given is 32pq2−4p2q + 12pq−7.
A constant is that term which does not any variable portion or which is fixed.
Here, all terms have variables in the form of p and q but The last term does not have any variable.
So, The terms which have variables are the terms which are not constant.
Hence, 32pq2, -4p2q and 12pq are not constant in the given expression.
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When given these two equations:
y = 2x + 4
y = 3x - 6,
How would I set up these systems of equations so that I could find the solution?
O Manipulate my variables to have the same coefficient, but different signs
O Make the two equations equal to each other and solve
O add the equations together
O subtract the equations together
There are 35 questions total that will be posted in mathematics as this is Algebra I!
Whoever answers correctly and with an explanation will receive Brainliest when the option shows up on my end!
PLEASE ANSWER ASAP!!
Answer:
subtract the equations
Step-by-step explanation:
the second answer option is nonsense.
if the equations really have exactly one solution, then it is impossible to make them equal.
if 2 equations can be made equal, that means they have infinitely many solutions.
the other answers have all some merit.
but the last one (subtract the equations) is the best one.
it means
y = 2x + 4
- y = 3x - 6
------------------
0 = -x + 10
after adding x to both sides we get
x = 10
with that we use any of the 2 basic equations to get y :
y = 2x + 4 = 2×10 + 4 = 24
this worked right away, because the y terms have the same coefficients (1). so, by subtracting one equation from the other we eliminate y as variable and can solve for x.
if that was not the case like in
2y = 4x + 8
y = 3x - 6
we would have to do some transformation or manipulation to bring the system into a form like here in the beginning.
and the first and the third answer option would apply.
a triangle has two sides of length 18 and 3. what is the largest possible whole-number length for the third side?
Answer:
20
Step-by-step explanation:
Triangle inequality says that the third side can only be 18-3...15, that is bigger than 15.
And 18+3... 21, that is smaller than 21.
If the third side is 21, the 18 and the 3 will just lay right on top of the 21 and not make a triangle. So it has to be 20 in order to be a whole number.
Which statement about the graph of y=12·(1/2)x is true?
The graph has an asymptote at y = 0.
Option (A) is correct.
What is the asymptote of the graph?
An asymptote is a straight line that constantly approaches a given curve but does not meet at an infinite distance. In other words, an Asymptote is a line that a curve approaches as it moves toward infinity.
The given function is:
[tex]y = 12.(\frac{1}{2})^x[/tex]
Now let's draw the graph for this given function,
By observing the graph we can conclude that the graph has an asymptote at y = 0 cause it is not meeting to the x-axis when it is approaching infinity.
Hence, the graph has an asymptote at y = 0.
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You deposit $3000 in an account earning 8% interest compounded monthly. How much will you have in the account in 5 years?
Answer: $5580
The formula for compound interest is:
A = P(1 + r/n)^nt
where A is the final amount, P is the initial principal (the amount you deposited in the account), r is the annual interest rate (8% in this case), n is the number of times the interest is compounded per year (12 times per year if the interest is compounded monthly), and t is the number of years the money is invested (5 years in this case).
Plugging in the values from the problem, we get:
A = $3000(1 + 0.08/12)^(12*5)
= $3000(1.0067)^60
= $3000*1.86
= $5580
So, after 5 years, you will have $5580 in the account if you deposit $3000 and earn 8% interest compounded monthly.
The value of amount after 5 years is,
⇒ A = $4,380
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
You deposit $3000 in an account earning 8% interest compounded monthly.
Hence, The value of amount after 5 years is,
⇒ A = 3000 [ 1 + 0.08]⁵
⇒ A = 3000 × 1.08⁵
⇒ A = 3000 × 1.46
⇒ A = $4,380
Thus, The value of amount after 5 years is,
⇒ A = $4,380
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A website displayed the following formula for the volume of a cone.
V = 1/3 nrh
James had never seen this formula before, but he knew there must be a typographical error in this formula. How could he tell
there was an error?
A. The calculated volume would be in linear units instead of cubic units.
B. The calculated volume would be in square units instead of cubic units.
C. The calculated volume would be in cubic units instead of square units.
D. The calculated volume would be in square units instead of linear units
James will declare the error by saying that the calculated volume would be in cubic units instead of square units.
What formula is correct for volume of cone?A cone has circular base and smooth curved surfaces that ends at the top.
volume of cone formula is denoted by V = 1/3×(πr²h) cubic unit
r is the radius of base of cone
h is the height of cone
π =3.142
hence, James needs to say that the unit of volume of cone should be written in cubic
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Which number represents a square root of 3 cos
3 [cos()+isin()]?
2
O √[cos()+sin()]
8
O
√3 cos (1) + i sin()]
5
O √3 [cos (STT)+isin (STT)]
6
O
3π
√3 [cos (3T) + sin(3)]
i
2
Square root of 3 cos
Consider the given data.
[tex]$$\begin{aligned}& \sin \theta+\sin ^2 \theta=1 \\& \sin \theta=1-\sin ^2 \theta \\& \sin \theta=\cos ^2 \theta\end{aligned}$$[/tex]
We have,
[tex]$$\begin{aligned}& =\cos ^{12} \theta+3 \cos ^{10} \theta+3 \cos ^8 \theta+\cos ^6 \theta+2 \cos ^4 \theta+2 \cos ^2 \theta-2 \\& =\cos ^{12} \theta+3 \cos ^{10} \theta+3 \cos ^8 \theta+\cos ^6 \theta+2\left(\cos ^4 \theta+\cos ^2 \theta-1\right) \\& =\left(\cos ^4 \theta+\cos ^2 \theta\right)^3+2\left(\cos ^4 \theta+\cos ^2 \theta-1\right)\end{aligned}[/tex]
[tex]$$Since, $\sin \theta=\cos ^2 \theta$Therefore,$$\begin{aligned}& =\left(\sin ^2 \theta+\sin \theta\right)^3+2\left(\sin ^2 \theta+\sin \theta-1\right) \\& =1^3+2(1-1) \\& =1+0\end{aligned}[/tex]
[tex]=1[/tex]
Hence, the required value is 1.
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Bryan says that every equation that has one variable has a solution to it. Disprove Bryan’s assertion with two counterexamples.
Bryan's assertion is not true. Here are two counter examples to disprove it:
1) x = 0
This equation has only one variable, x, but it has no solution because there is no value of x that will make the equation true.
2) x^2 + 1 = 0
This equation also has only one variable, x, but it has no solution because there is no value of x that will make the equation true.
These examples show that it is possible for an equation with one variable to have no solution.
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