The least count of Vernier calipers is 0.01cm and The length of the line is 10.23cm
Mean scale division:
The main scale is graduated with one division value equal to 1 mm. The length of the Vernier scale is the same as the n- scale of the Vernier scale and the (n-1) scale of the main scale.
Vernier Scale Division:
The minimum Vernier caliper number is also called the Vernier constant. Defined as the difference between the major and Vernier scales is represented mathematically as:
VC = 1 MSD – 1 VSD
If the Vernier scale has n divisions, it matches the (n-1) divisions of the main scale. Vernier Caliper :
LC = ( 1 - [tex]\frac{n-1}{n}[/tex]) MSD
According to the problem
1MSD= 1mm
MSR = 10.2cm
Here, 10VSD = 9MSD
1VSD = (9/10)MSD
=(9/10)×1mm
= (9/10)mm
Least count of Vernier calipers L.C = 1MSD - 1VSD
=1mm - (9/10)mm
=0.01mm
Therefore, Least Count (LC) = 0.01cm
Therefore,
Length of the line = MSR + (VSD × L.C)
= 10.2 + (3 × 0.01)
= 10.2+(0.03)p
= 10.23 cm
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Using Pythagoras' theorem, calculate the length of the hypotenuse in this right-angled triangle. Give your answer in centimetres (cm) to 1 d.p. 4.8 cm 2 cm Not drawn accurately
Answer: 5.2 cm
Step-by-step explanation:
4.8 squared + 2 squared is 23.04 + 4, which is 27.04. The square root of that is 5.2
Identify the surface with the given vector equation. r(s, t) = (s, t, t^2-s^2) elliptic cylinder circular paraboloid hyperbolic paraboloid plane circular cylinder
Therefore , the solution of the given problem of equation comes out to be x² + z²= y, a circular paraboloid.
What is equation?When a math formula employs the equals symbol (=), it appears to be a rule that connects two expression and denotes equality. An equation in algebra is a factual declaration that shows that several mathematical variables are all equal. For instance, the values ptdc + 6 and 12 in the equation obd + 6 = 12 have an equal sign. The link between the words on either side of each letter is described by a mathematical formula. The sentence and the insignia are frequently same.
Here,
The vector solution is r(s, t) = s, t, t² - s².
When comparing to r(s, t) = x, y, and z, x = s, y = t, and z = 2 - s²
So, z = y² - x²
, a hyperbolic paraboloid (2)
r(s, t) = s sin³t, s², s cos³t is the vector equation that is presented.
When comparing with withr(s, t) = x, y, and z,
x = ssin³t, y = s², and z = s* cos³t²
As a result, x2 + z2 = s2 (sin23t + cos23t),
sin23t + cos23t = 1,
x² + z² = s², s² = y, and x² + z²= y,
a circular paraboloid.
Therefore , the solution of the given problem of equation comes out to be x² + z²= y, a circular paraboloid.
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A rectangular field measures 63.9m by 104.6metres find the minimum number of poles to be Erected for fencing if they are to be at most 2.4meters apart.
The minimum number of poles to be Erected for fencing are 26.
What is the perimeter of a rectangle?The perimeter of a rectangle is -
P = 2{L + B}
Given is that a rectangular field measures 63.9 meters by 104.6 meters.
The perimeter of the rectangular field will be -
P = 2(63.9 + 104.6)
P = 2(168.5)
P = 337
Let the total number of poles that can be erected are {x}. Then, we can write that -
2.4x ≤ 63.9
x ≤ (63.9/2.4)
x ≤ 26.66
x = 26 {approx.}
Therefore, the minimum number of poles to be Erected for fencing are 26.
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Gertrude takes out a $5,500 subsidized stafford loan, which must be paid back in ten years. gertrude will graduate four years after taking out the loan. if the loan has an interest rate of 6.8%, compounded monthly, and gertrude makes monthly payments, how much interest will she pay by the time the loan is repaid? round all dollar values to the nearest cent. a. $4,462.40 b. $1,213.28 c. $1,713.69 d. $2,094.80 please select the best answer from the choices provided a b c d
$2094.28 interest will she pay by the time the loan is repaid.
What is compounded monthly?
Theoretically, continuously compounded interest means that an account balance continuously earns interest as well as reinvesting that interest into the balance so that it too earns interest.
As we know the formula of per month installments
[tex]E.M.I = \frac{P*r*(1+r)^n}{(1+r)^n^-^1}[/tex]
By putting the value of P (loan applied for)=$5500
r (Monthly rate of interest) [tex]= \frac{6.8}{12*100}[/tex]
n = number of monthly installments = 10 × 12 = 120
[tex]E.M.I = \frac{5500*(\frac{6.8}{12*100} )(1+\frac{6.8}{1200} )^1^2^0}{(1+\frac{6.8}{1200} )^1^2^0-1}\\ E.M.I= \frac{5500*0.00567*1.971}{1.971-1}\\ E.M.I = \frac{61.461}{0.971} \\E.M.I = 63.29[/tex]
E.M.I.=$63.29
Total Installments of loan =120
Therefore total amount paid against loan = $63.39 × 120 = 7594.28
So interest paid = 7594.28 - 5500= $2094.28
Hence, $2094.28 interest will she pay by the time the loan is repaid.
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Aume that y varie inveraly with x. If y equal 7 when x equal 2/3, find y when x =7/3
If y equals 7 when x equal 2/3, so when x =7/3 the y is equal to 14/7
If y varies inversely with x, this means that the product of x and y is a constant. So if we know that y = 7 when x = 2/3, then we can set up the equation: x*y = k, where k is the constant.
Substituting the known values we get: (2/3)*7 = k.
So we know that k = 14/3
Now, we can use this value of k to find the value of y when x = 7/3.
x*y = k
(7/3)*y = 14/3
y = 14/7
So, when x = 7/3, y = 14/7
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signment Active
Assignment
A'
Practice with rotations.
B'
C'
D
C
A
B
Examine the rotation. Which best describes point D?
O angle of rotation
O center of rotation
O image
O pre-image
Point D is the center of rotation in the given figure.
What are coordinates?A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.
Triangle A'B'C' rotating about the point d. Observe the Positional relation between ABC and A'B'C' you can get the result. There is no more information given Otherwise i will prove it to you theoretically.
Therefore, Point D can be described as Center of rotation.
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Complete question:
Help me with this question for BRAINLIEST
Last year, Christine had $20,000 to invest. She invested some of it in an account that paid 8% simple interest per year, and she invested the rest in an account that paid 10% simple interest per year. After one year, she received a total of $1920 in interest. How much did she invest in each account?
First account:
Second account:
Answer:
First account: $16,000
Second account: $4,000
Step-by-step explanation:
Let x be the amount Christine invested in the 8% account and y be the amount she invested in the 10% account. We know from the problem that:
x + y = $20,000 (the total amount she had to invest)
0.08x + 0.1y = $1920 (the total amount of interest she received)
We can use the first equation to solve for one variable in terms of the other.
x = $20,000 - y
Now we substitute this expression into the second equation:
0.08($20,000 - y) + 0.1y = $1920
Solving for y:
0.08x + 0.1y = $1920
0.08($20,000 - y) + 0.1y = $1920
1,600 - 0.08y + 0.1y = $1920
-0.08y = -320
y = $4,000
Now, we know that Christine invested $4,000 in the 10% account, we can use the first equation again to find out how much she invested in the 8% account:
x + y = $20,000
x + $4,000 = $20,000
x = $16,000
So Christine invested $16,000 in the 8% account and $4,000 in the 10% account
How do I do this help me please
Answer: x = 81
Step-by-step explanation:
Quite a straightforward question.
Given equation is 61+20=x
So you just need to add 61 and 20 to get the value of x,
x=81
0.25f = 10 please helpp
Answer:
f = 40
Step-by-step explanation:
0.25f = 10
f = 40
Let's check
0.25(40) = 10
10 = 10
So, f = 40 is the correct answer.
after a hailstorm, a large car dealership wants to determine the proportion of cars that have damage. the service department randomly selects 50 cars on the dealership lot, examines them, and determines that 18 have damage. assuming all conditions have been met, they construct a 99% confidence interval for the true proportion of cars with damage from the storm. what are the calculations for this interval?
On solving the provided question we can say that 50 vehicles, 11 of which are damaged, leave 39 intact. Conditions for inference are satisfied because both proportion number of successes and failures is greater than 10.
what is proportionality?Relationships that consistently have the same ratio are referred to as proportionate. For instance, the average quantity of apples per tree determines how many trees there are in an orchard and how many apples are in a harvest of apples. In mathematics, proportional denotes a linear relationship between two numbers or variables. The other amount doubles when the first quantity does. The other lowers as well when one of the variables falls to 1/100th of its prior value.
Yes, the prerequisites for inference are satisfied.
Inference requirements:
The sample must have at least 10 successes and 10 failures in order to construct a confidence interval for a population proportion.
50 vehicles, 11 of which are damaged, leave 39 intact.
Conditions for inference are satisfied because both the number of successes and failures is greater than 10.
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Simplify: 1. Write the prime factorization of the radicand. 2. Apply the product property of square roots. Write the radicand as a product, forming as many perfect square roots as possible.
The prime factorization of the radicand 2 is 9√15.
A number can be expressed as the product of its prime components through the process of prime factorization. A number with precisely two elements, 1 and the number itself, is said to be a prime number.
As an illustration, let's use the number 30. We know that 30 = 5 × 6, but 6 is not a prime number. The number 6 can further be factorized as 2 × 3, where 2 and 3 are prime numbers. Therefore, the prime factorization of 30 = 2 × 3 × 5, where all the factors are prime numbers.
Given that,
x= 3√135
Solving the equation further using the rule √A*B = √A*√B
x= 3√9*15
x= 3√9*√15
x=3*3*√15
x= 9√15
Therefore, the prime factorization of the radicand 2 is 9√15.
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1. Kevin regularly travels from Manchester to Oxford.
He travels on two different trains.
His first train is from Manchester to Birmingham and his second train is from Birmingham to
Oxford.
On the 1st train, the probability that a seat has a table is 3/5
On the 2nd train, the probability that a seat has a table is
3/10
The 1st train has a total of 400 seats.
(a) Work out an estimate for the number of these 400 seats that have a table.
Answer:
240 seats with tables
Step-by-step explanation:
Let's start by making our fraction out of 10 on the first train to make it easier on us, this fraction would be [tex]\frac{6}{10}[/tex]. Then, to get the number of seats on this train that would have a table, we need to multiply 400 by our fraction, giving us:
[tex]400 * \frac{6}{10} = 2400/10 = 240[/tex]
So, the 1st train should have approximately 240 seats with tables.
Hope this helped!
select the correct answer. it is estimated that approximately one-half of all aluminum cans will be recycled each year. if a soft drink company produces 350,000 cans one year, how many cans are still in use after 4 years of recycling and re-using, using this function? nt
After 4 years of recycling and re-using, approximately average one-half of the original 350,000 cans produced by the soft drink company would still be in use.
1. 350,000 cans produced in 1 year
2. 50% of cans recycled each year
3. 50% of 350,000 cans = 175,000 cans
4. After 4 years, 175,000 cans still in use.
Each year, it is predicted that about half of all aluminum cans will be recycled. This means that if a soft drink company produces 350,000 cans one year, then after four years, approximately one-half of the original cans produced would still be in use. This can be calculated by taking the initial number of cans produced (350,000), multiplying it by fifty percent (50%), and then multiplying that number by four (4). The final result is that after four years of recycling and re-using, approximately 175,000 cans would still be in use.
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Evaluate: 2−1 × 23 /24
Answer:
0.9583
Step-by-step explanation:
(2-1) * (23÷24)
1*0.983
=0.9583
Answer:
[tex]\frac{23}{24} = 0.95[/tex]
Step-by-step explanation:
According to BODMAS
What is the volume of the cylinder? Round to the nearest hundredth and approximate using π = 3.14.
cylinder with a segment from one point on the circular base to another point on the base through the center labeled 2.6 feet and a height labeled 4.4 feet
23.35 cubic feet
35.92 cubic feet
71.84 cubic feet
93.4 cubic feet
Answer:
23.35 cubic feet
Step-by-step explanation:
The volume of a cylinder can be calculated using the formula: V = π * r^2 * h, where r is the radius of the circular base, h is the height of the cylinder.
To find the radius, we use the formula for the circumference of a circle: C = 2πr
The circumference of the circular base is 2.6 feet, so we can set this equal to 2πr and solve for r: 2.6 = 2πr, r = (2.6) / (2π) = 0.816
Now that we know the radius, we can substitute it into the volume formula: V = π * (0.816)^2 * 4.4 = 23.35 cubic feet
So the answer is 23.35 cubic feet, which closest to A. 23.35 cubic feet
pls help super simple !!
The values of the expression are given by the inequality below:
11/3 < 5/x + 2 < 9/2
How to find the possible values of the expression?Here we want to find the possible values of the expression:
5/x + 2
Where we have the restriction:
2 < x < 3
Notice that x is on the denominator, so when x takes the largest value x = 3, we will have a lower bound for the expression:
5/3 + 2 = 5/3 + 6/3 = 11/3
When x takes the smallest value, we will get the upper bound:
5/2 + 2 = 5/2 + 4/2 = 9/2
Then the possible values of the expression are:
11/3 < 5/x + 2 < 9/2
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The volume of a rectangular prism is given by the expression10x3 + 46x2 – 21x – 27. The area of the base of the prism is given by the expression 2x2 + 8x – 9. Which of the following expressions represents the height of the prism? (V = Bh)
8x - 3
3x - 5
5x + 3
42x + 3
The height of the prism is 5x + 3 units.
What is volume?
A measurement of three-dimensional space is volume. It is frequently expressed numerically using SI-derived units, as well as different imperial or US-standard units. Volume and the definition of length are related.
Given:
The volume of a rectangular prism is given by the expression
10x^3 + 46x^2 – 21x – 27. The area of the base of the prism is given by the expression 2x^2 + 8x – 9.
We have to find the height of prism.
Volume of the rectangular prism = Base × Height
The expression is in the Question be
10x ³ + 46 x² - 21x -27
And the area of the base of the prism is given by the expression
2x² + 8x - 9 .
Put in the formula
10x ³ + 46 x² - 21x -27 = 2x² + 8x - 9 × Height
The factor of 10x ³ + 46 x² - 21x -27 are (5x +3 )(2x² + 8x - 9) .
put in the formula
(5x +3 )(2x² + 8x - 9) = (2x² + 8x - 9) × Height
Cancelled 2x² + 8x - 9 on both side.
(5x+3)unit = Height
Hence, the height of the prism is 5x + 3 units.
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according to westgard and clia, how many specimens should be run by each method on the same day for 8 to 20 days to compare a test method with a comparative method?
According to Westgard and CLIA , 40-100 specimens should be run by each method on the same day for 8 to 20 days to compare a test method with a comparative method.
CLIA stands for Clinical Laboratory Improvement Amendments . The regulations under CLIA includes, the federal standards which is applicable to all U.S. facilities or sites which are test human specimens for health assessment or it is used to diagnose, prevent, or treat disease.
The Federal Register is the official daily publication for federal agencies and the organization's notices, proposed regulations, and final rules, as well as executive orders and other presidential papers. The regulations published in the Code of Federal Regulations and enforced by federal agencies are known as final rules.
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How do you this help please
Answer:
Step-by-step explanation:
The height of her pear tree is 20 inches
First subtract 46 from 26 to get 20. So 20+26=46inches
x=20
Step-by-step explanation:
you know how to transform an equation ?
you need to apply the same operating to both sides of the equation. always. otherwise the equality relation is destroyed.
x + 26 = 46
to get to "x = ..." the "+ 26" is in the way.
clearly we need to subtract 26. but we need to do it on both sides.
x + 26 - 26 = 46 - 26
x = 20
that simply means the last tree is 20 in.
it is 26 in shorter than 46 in (fig tree).
that's all there is to it.
Sorry, but may I ask what the answer to this question is?
Which is not a combination that can be made if Joe has a pair of yellow, orange, black, and blue shoes with yellow, red, and brown shoelaces?
A. orange shoes with red shoelaces
B. black shoes with yellow shoelaces
C. blue shoes with brown shoelaces
D. yellow shoes with black shoelaces
Answer:
D
Step-by-step explanation:
There are no black shoelaces
Add and Subtract Fractions Quiz
Select the correct solution for the expression.
2/5+3/8
A. 2/5+3/8=5/13
B. 16/40+15/40=31/40
C. 10/40+24/40=34/40
D. 2/5+3/8=6/40
Answer:
B.
Step-by-step explanation:
You need to find a common denominator. In this case, 40. Convert 2/5 to 16/40 and 3/8 to 15/40. Add the numerators (16 + 15) together. The denominator stays the same.
Suppose you want to have 800,000 for retirement in 35 years your account earns 9% interest how much would you need to deposit in the account each month
Therefore , the solution of the given problem of interest rate comes out to be $271.94 you will be depositing every month.
What is interest, exactly?The original capital is multiplied by the interest rate, the loan term, and other factors to calculate simple interest. A simple return is equal to the principal plus interest hrs. This method of calculating interest is the simplest. The most typical method is to calculate it as a fraction of the principal amount. For instance, he was only liable for paying his portion of the interest if he acquired $100 from a buddy and agreed to repay the money on 5% interest. $x (0.05) = $5. When you make loans or lend money, you both have to pay interest. Interest is frequently determined as a percentage of a loan.
Here,
Given : interest = 9%
time =35 years
amount = 800000
Using formula :
=> FV.( r / [tex](1+r) ^{n}[/tex] -1)
=> 800000 (0.09/12 / [tex](1 + 0.09/12 )^{12*35}[/tex] - 1 )
=> 271.94
Therefore,
the total amount you deposit will only be
=> 35*12 times $271.94, or
roughly total amount => 35*12*271.94= 114214.8 dollars, out of the estimated $800,000.
The balance will be what the account earns or accumulates over the next 35 years.
Therefore , the solution of the given problem of interest rate comes out to be $271.94 you will be depositing every month.
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In parallelogram ABCD, AB = 14cm. The altitude corresponding to AB is 6 cm and the altitude corresponding to BC is 7 cm. Find AD.
If in parallelogram ABCD, AB = 14cm. The altitude corresponding to AB is 6 cm and the altitude corresponding to BC is 7 cm. The AD is 8.57.
How to find AD?Given data:
AB= 14cm
Altitude corresponding AB =6cm
Altitude corresponding to BC =7cm
So,
1/2 × AB × altitude 1 = 1/2 × AD × altitude 2
AB × altitude 1 =AD × Altitude 2
10 × 6 = AD × 7
60 = AD × 7
AD = 60/7
AD = 8.57
Therefore we can conclude that AD is 8.57.
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my sock drawer is always a mess. at the beginning of the week, i have 4 pairs of black socks, 5 pairs of blue socks, 3 pairs of brown socks, and 2 pairs of multi-colored socks. every morning, i grab a pair and hope for the best. if on the 1st three days i pick 1 black pair and 2 blue pairs, what is the probability that i'll pick a solid color on the 4th day?
The probability of picking a solid color on the 4th day is 8/10 = 4/5.
The probability of picking a solid color on the 4th day is calculated by taking the total number of solid color pairs remaining in the drawer and dividing by the total number of pairs remaining.
On the first three days, you picked 1 black pair and 2 blue pairs, so you have 3 black pairs, 3 blue pairs, 3 brown pairs and 2 multi-colored pairs left in your drawer.
The total number of solid color pairs remaining is 8, and the total number of pairs remaining is 8+2 = 10.
So, the probability of picking a solid color on the 4th day is 8/10 = 4/5.
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amy was born on a tuesday. what is the probability that exactly two of her three best friends were also born on tuesday? express your answer as a common fraction.
The probability of being born on a Tuesday is= 18/343
What is probability ?The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true.
An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
The likelihood that an event will occur increases with its probability.
A straightforward illustration is tossing a fair (impartial) coin.
The coin is fair, thus the outcomes "heads" and "tails" are equally likely; the likelihood of "heads" is equal to the likelihood of "tails"; and because there are no other conceivable outcomes, the likelihood of either "heads" or "tails" is 1/2 (which is also an acceptable spelling).
Hence, The probability of being born on a Tuesday is= 18/343.
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State the fifth and seventh terms of the sequence -2, -3, -4½..
Answer:
see explanation
Step-by-step explanation:
there isa common difference between consecutive terms, that is
[tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{-3}{-2}[/tex] = [tex]\frac{3}{2}[/tex]
[tex]\frac{a_{3} }{a_{2} }[/tex] = [tex]\frac{-4\frac{1}{2} }{-3}[/tex] = [tex]\frac{3}{2}[/tex]
this indicates the sequence is geometric with nth term
[tex]a_{n}[/tex] = a₁ [tex](r)^{n-1}[/tex]
where a₁ is the first term and r the common ratio
here a₁ = - 2 and r = [tex]\frac{3}{2}[/tex] , then
[tex]a_{n}[/tex] = - 2 [tex](\frac{3}{2}) ^{n-1}[/tex] , thus
a₅ = - 2 ([tex](\frac{3}{2}) ^{4}[/tex] = - 2 × [tex]\frac{81}{16}[/tex] = - [tex]\frac{81}{8}[/tex]
and
a₇ = - 2 [tex](\frac{3}{2}) ^{6}[/tex] = - 2 × [tex]\frac{72 9}{64}[/tex] = - [tex]\frac{729}{32}[/tex]
Mr. Morris is going to save money and replace his sailboat's mainsail himself. He must determine the area of the mainsail in order to buy the correct amount of material. Calculate the area of the parallelogram to determine how much material should be purchased. Be sure to explain how to decompose this shape into rectangles and triangles. Describe their dimensions and show your work. Parallelogram with base of 20 feet, height of 15 feet, and triangular base of 4 feet.
According to the given information the area of the parallelogram is 300ft².
How do parallelograms function?A geometric shape having parallel sides in two dimensions is called a parallelogram. It is a type of four-sided polygon where each parallel set of sides is the same length (commonly referred to as a quadrilateral). The adjacent angles of such a parallelogram add up approximately 180 degrees.
Width of parallelogram = 20 feet.
Base of triangle = 4 feet.
Height of triangle = 15 feet.
To figure out the parallelogram's surface area:
To begin with, we would calculate the triangles' surface areas.
Note: The parallelogram supplied can be divided into two (2) triangles.
the triangle's surface area formula.
The formula: yields the triangle's area mathematically.
[tex]\begin{matrix}\mathrm{\ Area\ }=\frac{1}{2}\times\mathrm{\ base\ } \times\mathrm{\ height\ } \\\mathrm{\ Area\ }=\frac{1}{2}\times4\times15\\\mathrm{\ Area\ }=2\times15\\\mathrm{\ Area\ }=\mathbf{30}f^2\\\end{matrix}[/tex]
For the two (2) triangles:
[tex]Area =2\times30\\Area =\mathbf{60}ft^2[/tex]
For the rectangle left:
[tex]Length =15ft.\\Width =20-4=16ft.\\Area = length \times width\\Area =15\times16\\Area =\mathbf{240}ft^2[/tex]
Now, the area of the parallelogram:
Area of parallelogram =60+240
Area of parallelogram [tex]=\mathbf{300}{\rm ft}^2[/tex]
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the region inside the cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ)
The region inside cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ) is equal to [tex]\frac{\pi}{4}[/tex]
Now, According to the question:
We will first draw both of them on the same plane and then find their point of intersection. Then using the given information, we will shade the region whose area we have to find.
The point of intersection of r = 1 + cos(θ) and r = 3 cos(θ)
1 + cos(θ) = 3 cos(θ)
1 = 2 cos(θ)
=> cos(θ) = 1/2
=> θ = [tex]cos^-^1[/tex][tex]\frac{1}{2}[/tex]
=> θ = ±[tex]\frac{\pi }{3}[/tex]
So, two curves intersect at θ = ± [tex]\frac{\pi }{3}[/tex]
Area of the cardioid,
[tex]A_1 = \int\limits^\pi _\frac{\pi }{3} {\frac{1}{2}(1+cos\theta)^2 } \, d\theta= \frac{1}{2} \int\limits^\pi _\frac{\pi }{3} (1+cos^2\theta+2cos\theta) } \, d\theta=\frac{1}{2}[/tex]
[tex]\int\limits^\pi _\frac{\pi }{3} {(1 + \frac{cos2\theta+1}{2} +2cos\theta)} \, d\theta[/tex]
=> [tex]A_1 = \frac{1}{2}[\theta+\frac{1}{2}(\frac{sin2\theta}{2} +\theta)+2sin\theta ]^\pi _\frac{\pi }{3}[/tex]
=> [tex]A_1= \frac{1}{2}[\frac{3\pi }{2}+0+0-\frac{\pi }{2}-\frac{\sqrt{3} }{8} -\sqrt{3} ][/tex]
=> [tex]A_1= \frac{1}{2} [\pi +\frac{-9\sqrt{3} }{8} ]\\\\A_1 = \frac{\pi }{2} - \frac{9\sqrt{3} }{16}[/tex]
Area of the circle,
[tex]A_2 = \int\limits^\frac{\pi }{2} _\frac{\pi }{3} {\frac{1}{2}(3cos\theta)^2 } \, d\theta \\\\A_2 = \frac{9}{2} \int\limits^\frac{\pi }{2} _\frac{\pi }{3} \frac{cos2\theta+1}{2}d\theta\\ \\A_2 = \frac{9}{4}[\frac{sin2\theta}{2}+\theta ]^\frac{\pi }{2}_\frac{\pi }{3}[/tex]
[tex]A_2 = \frac{9}{4}[0+\frac{\pi }{2} -\frac{\sqrt{3} }{4} -\frac{\pi }{3} ][/tex]
[tex]A_2=\frac{9}{4}[\frac{\pi}{6}-\frac{\sqrt{3} }{4} ][/tex]
[tex]A_2= \frac{3\pi }{8}-\frac{9\sqrt{3} }{16}[/tex]
Area of the shaded region,
[tex]A = A_1-A_2[/tex]
[tex]A = \frac{\pi }{2} - \frac{9\sqrt{3} }{16} - \frac{3\pi }{8}+\frac{9\sqrt{3} }{16}[/tex]
[tex]A = \frac{\pi }{8}[/tex]
This is the area of the shaded region in the first and the second quadrant. We see that by symmetry, the area of the shaded region in the first and the second quadrant is equal to the area of the shaded region in the third and the fourth quadrant.
So, the total area is 2 ×[tex]\frac{\pi}{8}[/tex] = [tex]\frac{\pi}{4}[/tex]
Hence, the region inside cardioid r = 1 + cos(θ) and outside the circle r = 3 cos(θ) is equal to [tex]\frac{\pi}{4}[/tex]
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Kevin is travelling from Manchester to Oxford tomorrow.
(b) Complete the probability tree diagram to show the outcomes of Kevin's seating on the two
trains.
Label clearly the branches of the probability tree diagram.
The probability tree diagram has been started in the space below.
315
Seat on the
first train
with a
table
without
a table
Seat on the
second train
Answer:
I don't know the answer to this question
The inverse of f(x) would be represented by:
f(x)
ƒ¹(x)
f(x)-¹
fog(x)
None of the choices are correct.
If g(x) is the inverse function of f(x) and[tex]$f^{\prime}(x)=\frac{1}{1+x^4}$[/tex], then [tex]$g^{\prime}(x)$[/tex] is [tex]1+[\mathrm{g}(\mathrm{x})]^4\end{aligned}[/tex]
What is Inverse function?By applying the formula x=-b/2a to find the quadratic's vertex, the result can then be used to replace y in the original equation. Substitute the vertex into the equation y=a(x-h)2+k in the vertex form. (A will not change; h is x; and K is y.) It is referred to as being in standard form when the quadratic function f(x) = a(x - h)2 + k is not equal to zero. The graph opens either upward or downward depending on whether an is positive or negative. The vertex is the point, while the vertical line x = h is the line of symmetry (h,k).
Correct option is A)
[tex]& \mathrm{g}=\mathrm{f}^{-1} \\[/tex]
[tex]& \mathrm{f}(\mathrm{g}(\mathrm{x}))=\mathrm{x}[/tex]
Differentiate w.r.t.x
[tex]& \mathrm{f}^{\prime}(\mathrm{g}(\mathrm{x})) \cdot \mathrm{g}^{\prime}(\mathrm{x})=1 \\[/tex]
[tex]& \therefore \frac{1}{1+(\mathrm{g}(\mathrm{x}))^4} \cdot \mathrm{g}^{\prime}(\mathrm{x})=1 \\[/tex]
[tex]& \mathrm{~g}^{\prime}(\mathrm{x})=1+[\mathrm{g}(\mathrm{x})]^4\end{aligned}[/tex]
The complete question is,
If [tex]$\mathrm{g}(\mathrm{x})$[/tex]is the inverse function of [tex]$\mathrm{f}(\mathrm{x})$[/tex] and [tex]$\mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{1+\mathrm{x}^4}$[/tex], then [tex]$\mathrm{g}^{\prime}(\mathrm{x})$[/tex]is
A [tex]$1+[\mathrm{g}(\mathrm{x})]^4$[/tex]
B [tex]$1-[g(x)]^4$[/tex]
C [tex]$1+[\mathrm{f}(\mathrm{x})]^4$[/tex]
D [tex]$\frac{1}{1+[\mathrm{g}(\mathrm{x})]^4}$[/tex]
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