The point among the answer choices which is not in the solution set of the inequality is; (-1, 3).
Which point is not included in the solution set of the inequality?It follows from the task content that the vertex of the two-variable inequality in discuss is; (1, 2).
On this note, it follows that; the two variable inequality described by a parabola opening upward can be written by means of its vertex as;
y >= (x - (1))² + 2
y >= (x-1)² + 2.
By testing points among the answer choices, the point which is not part of the solution set is; (-1,3).
This follows from the fact that; at x=-1;
The value of y is;
y >= (-1-1)² +2
y >= 6
Hence, since the value of y= 3 which is less than, 6, it follows that the point (-1,3) is not in the solution set.
Remarks: The answer choices in the complete question are as follows;
(–1, 3)
(0, 4)
(1, 5)
(2, 4)
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Marin corporation had a projected benefit obligation of $3235000 and plan assets of $3474000 at January 1, 2020 . Marin also had a net acturial loss of $505740 in accumulated OCI at January 1, 2020.The average remaining service period of Marin's employees is 7.80 years . Compute Marin's minimum amortization of the actuarial loss. Minimum amortization of the actuarial loss
the minimum amortization is given as 20300 dollars
How to solve for the amortizationWe have the value of A to be $3235000
while we have the value of B to be $3474000
Of these two values the greatest or the highest is that of the option B.
Next we have to find the corridor value using 10 percent
0.10 * 3474000
= 347400
$505740 - 347400
= 158340
The number of years = 7.8
minimum amortization = 158340/7.8
= 20300 dollars
Hence the minimum amortization is given as 20300 dollars
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Find the cube root of 1+¡^5
Volumes of the solids with disk or shell method?
The answer to the questions of volumes are given as follows
a) [tex]v=128 \pi[/tex]
b) [tex]v=\frac{128}{3} \pi[/tex]
c)[tex]v=\frac{1024}{5} \pi[/tex]
d)[tex]V=\frac{13568}{15} \pi[/tex]
Generally, the questions are mathematically solved below
[tex]y=\sqrt{x}, y=4, x=0[/tex]
a) x-axis
if $y=4, x=16, x=0$
Using disk method
[tex]} v=\pi \int_{0}^{16}(\sqrt{x})^{2} d x \\[/tex]
[tex]v=\pi\left(\frac{x^{2}}{2}\right)_{0}^{16} \\[/tex]
[tex]v=\frac{\pi}{2} \times 16 \times 16 \\[/tex]
[tex]v=128 \pi[/tex]
b) line y=4
if x=0, y=0 ;
[tex]y=\sqrt{x} \Rightarrow x=y^{2}[/tex]
Using shell method
[tex]v = \int_{0}^{4} 2 \pi(4-y) \cdot y^{2} d y \\[/tex]
[tex]v=2 \pi \int_{0}^{4}\left(4 y^{2}-y^{3}\right) d y[/tex]
[tex]v=2 \pi\left[\frac{4 y^{3}}{3}-\frac{y^{4}}{4}\right]_{0}^{4} \\[/tex]
[tex]v=\frac{2 \pi}{12}[1024-768] \\[/tex]
[tex]v=\frac{512 \pi}{12} \\[/tex]
[tex]v=\frac{128}{3} \pi[/tex]
c) y-axis
0 ≤ y ≤ 4
x=y^2
Using disk method
volume
[tex]v=\pi \int_{0}^{4} y^{4} d y$[/tex]
[tex]v=\pi\left(\frac{y 5}{5}\right)_{0}^{4} \\[/tex]
[tex]v=\frac{1024}{5} \pi[/tex]
d) line x=-1
y=√x, y=4, x=0
0 ≤ x ≤ 6
Using shell method
volume is
[tex]V=\int_{0}^{16} 2 \pi(1+x) \sqrt{x} d x$[/tex]
[tex]V=2 \pi\int_{0}^{16}(x^{1 / 2}+x^{3 / 2}\right) )d x\right. \\[/tex]
[tex]V=2 \pi\left[\frac{x^{3 / 2}}{3 / 2}+\frac{x^{5 / 2}}{5 / 2}\right]_{0}^{16} \\[/tex]
[tex]V=2 \pi\left[2 / 3 \cdot\left(4^{2}\right)^{3 / 2}+2 / 5\left(4^{2}\right)^{5 / 2}\right] \\[/tex]
[tex]V=4 \pi / 3 \cdot 4^{3}+4 \pi / 5 \cdot 4^{5} \\[/tex]
[tex]V=4 \pi\left(\frac{1}{3}+\frac{4^{2}}{5}\right)[/tex]
[tex]V=\frac{256}{15}(5+48) \pi \\[/tex]
[tex]V=\frac{256 \times 53}{15} \pi \\[/tex]
[tex]V=\frac{13568}{15} \pi[/tex]
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What is the direction and magnitude of the following correlation coefficients
a. -0.81
b. 0.40
c. 0.15
d. -0.08
e. 0.29
-0.81 has negative direction and o.81 is the magnitude, 0.40 has positive direction and 0.40 is the magnitude, 0.15 positive direction and 0.15 is the magnitude, -0.08 has negative direction and 0.08 is the magnitude and 0.29 has positive direction and 0.29 is the magnitude
What is Vector?A quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another.
-0.81
Minus zero point eight one has negative direction and o.81 is the magnitude
0.40
zero point four zero has positive direction and 0.40 is the magnitude
0.15
Zero point one five has positive direction and 0.15 is the magnitude
-0.08
Minus zero point zero eight has negative direction and 0.08 is the magnitude
0.29
Zero point two nine has positive direction and 0.29 is the magnitude
Hence -0.81 has negative negative direction and o.81 is the magnitude, 0.40 has positive direction and 0.40 is the magnitude, 0.15 positive direction and 0.15 is the magnitude, -0.08 has negative direction and 0.08 is the magnitude and 0.29 has positive direction and 0.29 is the magnitude
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Which choice is equivalent to the expression below?
√6 +2√√3+√27-√12
A. 53-6
OB. 2√3-√21
OC. 3√3+√6
OD. 5√3
The Evaluating expression √6 +√6+√27-√12 become √3(2√2+ 1) after simplification.
According to the statement
We have given that one evaluating expression which is √6 +√6+√27-√12
And we have to simplify this expression by evaluating it.
So, Given expression is:
√6 +√6+√27-√12
√6 +√6+√(9*3)-√(4*3)
√6 +√6+3√(3)-2√(3)
√6 +√6+√3( 3-2)
After evaluating the expression it become
√6 +√6+√3(1)
√(3*2) +√(3*2) +(1)√3
Take common from above expression then
√3(√2 +√2 ) +√3
√3(2√2) +√3
√3(2√2+ 1)
Now the expression √6 +√6+√27-√12 become √3(2√2+ 1) after simplification.
So, The Evaluating expression √6 +√6+√27-√12 become √3(2√2+ 1) after simplification.
Disclaimer: This question was incomplete. Please find the full content below.
Question:
Which choice is equivalent to the expression below?
√6 +√6+√27-√12
A. √3(2√2+ 1)
B. 2√3-√21
C. 3√3+√6
D. 5√3
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tammy smith deposits $5,000 in First Internet Bank's 5-year CD, which pays 5.22% compounded monthly.How much will she have in the account at the end of 5 Years? How much interest did she earn?
Answer: a) $ 6,487.47
b) $ 1,487.47
Step-by-step explanation:
9. Raju sells a watch at 5% profit. Had he sold it for 24 more he would have gained 11%. Find the cost price of the watch.
Answer:
400$
Step-by-step explanation:
Let C.P. of the watch = Rs. 100
When profit =5%; S.P. = Rs. (100+5)
= Rs. 105
and when profit = 11%;
S.P. = Rs. (100+11)
=Rs.111
Difference of two selling prices
= Rs. 111-Rs.105 = Rs.6
When watch sold for Rs. 6 more; then C.P. of the watch = Rs.
100
6
When watch sold for Rs. 24 more; then C.P. of the watch = Rs.
100
6
×
24
= Rs.
100
×
24
6
=400
Identify the lateral area and the surface area of a cube with edge length 15 in.
The Lateral area of the cube = 900 in.²
The Surface area of the cube = 1,350 in.².
What is the Lateral Area of a Cube?A cube's lateral area is the area covered by the lateral surfaces of the cube shape. The formula for the lateral area of a cube is given as: LA = 4a², where:
a = edge length of cube.
What is the Surface Area of a Cube?The surface area of a cube is the total area of all its 6 surfaces. The formula for finding the surface area of a cube is: 6a², where:
a = edge length of cube.
Edge length of cube (a) = 15 in.
Lateral area = 4a² = 4(15²)
Lateral area = 4(225)
Lateral area of the cube = 900 in.²
Surface area = 6a² = 6(15²)
Surface area = 6(225)
Surface area of the cube = 1,350 in.².
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What’s the next 3 numbers 0 10 1011 1031 102113 10311213
The next three numbers here would be 0, 10, 1011, 1031, 102113, 10311213, 10411223, 1031221314, 1041222314.
What is a number sequence
The term number sequence is used to refer to the list of numbers that have the relationship of having a linked rule. The number that follows in the sequence has to be worked out from the previous numbers in that particular sequence. You have to follow the particular rule to be able to work out the number that has to follow in the sequence.
For example we have these numbers 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34 etc. We would have to understand the pattern of numbers that are in the sequence. We can see that there is a difference of 3 between the number that succeeds the previous one in the sequence.
How to get the next values in the sequenceThe first value here is given to be 0
Next we have the 1 0 = 10
The next that we have is 1 0 and 1 1 written as 1011
Next is 1 zero and three one = 1031
We continue to follow the pattern till we arrive at 1041222314. This is where the sequence would then have to stop.
Hence we have to conclude that the next numbers are 10411223, 1031221314, 1041222314.
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Solve the proportion: 3/2 = 4/p + 9
The answer is p = -8/15
Proportion can simply be defined as equating two ratios
The proportion of 3/2 = 4/p +9 can be calculated as follows
3/2 - 9/1 = 4/p
-15/2 = 4/p
cross multiply both sides together to find the value of p
-15×p= 4×2
-15p= 8
p= -8/15
Hence the answer is p= -8/15
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6 A car covers 162 km in 3 hrs 36min, Find the average speed of a car in km/hr?
Answer:
[tex]\huge\boxed{\sf v = 45 \ km/hr}[/tex]
Step-by-step explanation:
Given Data:Distance = S = 162 km
Time = t = 3hr 36/60 min = 3 hr 0.6 hr = 3.6 hr
Required:Average Speed = v = ?
Formula:[tex]\displaystyle v = \frac{S}{t}[/tex]
Solution:Put the givens in the above formula
[tex]\displaystyle v = \frac{162}{3.6} \\\\v = 45 \ km/hr\\\\\rule[225]{225}{2}[/tex]
Line A passes through the points (-8, 5) and (-5, 4). Line B passes through the points (0, 1) and (4, -1). Which of the following describes the relationship between line A an line B?
1. Lines A and B are parallel, because they have opposite reciprocal slopes.
2. Lines A and B are parallel, because they have the same slope
3. Lines A and B intersect, because their slopes have no relationship.
4. Lines A and B are perpendicular, because they have opposite reciprocal slopes.
[tex]\stackrel{\textit{\LARGE Line A}}{(\stackrel{x_1}{-8}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{-5}~,~\stackrel{y_2}{4})} ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{4}-\stackrel{y1}{5}}}{\underset{run} {\underset{x_2}{-5}-\underset{x_1}{(-8)}}} \implies \cfrac{4 -5}{-5 +8}\implies -\cfrac{1}{3} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{\LARGE Line B}}{(\stackrel{x_1}{0}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-1})} ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-1}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{4}-\underset{x_1}{0}}} \implies \cfrac{-1 -1}{4 +0}\implies -\cfrac{1}{2}[/tex]
keeping in mind that perpendicular lines have negative reciprocal slopes, and that parallel lines have equal slopes, well, those two slopes above aren't either, so since they're neither, and they're different, that means that lines A and B intersect.
A number cube is rolled three times. An outcome is represented by a string of the sort OEE (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). The outcomes are listed in the table below. Note that each outcome has the same probability.
The probability of occurrence for the events A, B and C is; 1/4.
What is the probability of occurrence of.the described events?For the first event A in which case, there's no odd number on the first two rolls, the possible events are; EEE and EEO. Consequently, the required probability is;
Event A = 2/8 = 1/4.
For the event B in which case, there's an even number on both the first and last rolls; the possible events are; EEE and EOE. Consequently, the required probability is;
Event B = 2/8 = 1/4.
For the event C in which case, there's an odd number on each of the first two rolls; the possible events are; OOO and OOE. Consequently, the required probability is;
Event C = 2/8 = 1/4.
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How is a marketing -oriented firm different from a production-oriented firm or a sales-oriented firm?
Marketing-oriented firms focus more on what the market needs above every other aspects.
What is a Marketing-oriented Firm?Marketing-oriented firms can be described as firms that major in identifying the needs and wants of customers and then creating specific products that are designed to meet the wants of such customers. One of the important elements of marketing-oriented firms is to ensure there is a demand for whatever products or services they offer.
Some examples of marketing-oriented firms are:
AppleCoca-ColaAmazonWhat is a Product-oriented Firm or a Sales-oriented Firm?A product-oriented firm focus more resources on and focus on researching and developing quality products rather than focusing more on the market to discover the wants of customers.
Sales-oriented firm tend to focus more on developing its sales force that will promote their products and services.
How Marketing-Oriented Firms are Different from the Product-oriented or Sales-oriented Firms?Marketing-oriented firms stand out and are different from the other two types of firms because they focus more on what the market needs above every other aspects.
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The volume of a rectangular box is 3x(3x + 4)(3x - 1). (Drawing is not to
scale.)
Length 3x + 4
Width 3x
Height 3x - 1
Which statement about the volume of the box is tru?
The volume of the rectangular box is the product of the base area and the height 3x - 1
How to determine the true statement?The given parameters are:
Length = 3x + 4
Width = 3x
Height = 3x - 1
The base area of the rectangular box is calculated as
Base area = Length * Width
Substitute the known values in the above equation
Base area = 3x(3x + 4)
This means that the volume of the rectangular box is the product of the base area and the height 3x - 1
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how to construct frequency table with sturge approximation
The stages to utilize to construct a frequency distribution table utilizing Sturge's approximation are as follows:
How to construct a frequency distribution table?The steps to create a frequency distribution table utilizing Sturge's approximation exist as follows;
Step 1: Estimate the range of the data:
This exists simply by finding the difference between the biggest and the least values.
Step 2: Decide on the inaccurate number of classes in which the provided data exist to be grouped. The formula for this is;
K = 1 + 3.322logN
where; K= Number of classes
logN = Logarithm of the whole number of observations.
Step 3: Specify the approximate class interval size:
This exists acquired by splitting the range of data by the number of classes and exists symbolized by h class interval size.
Step 4: Find the starting point:
The lower class limit should take care of the least value in the raw data.
Step 5: Determine the remaining class boundaries:
When you have reached the lowest class boundary, then you can count the class interval size to the lower class boundary to obtain the upper-class boundary.
Step 6: Circulate the data into separate classes.
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For a segment of a radio show, a disc jockey can play 7 records. If there are 13 records to select from, in how many ways can the program for this segment be arranged?
The number of ways the program can be arranged for this segment is 8, 648, 640 ways
What is permutation?It is important to note that the formula for finding the permutation or arrangement of a set of dats is given as;
Permutation = [tex]\frac{n!}{(n-r)!}[/tex]
where
n = number to select from = 13r = number of objects selected = 7Let's substitute the values into the formula for permutation, we have
Permutation = [tex]\frac{13!}{13 - 7!}[/tex]
Find the difference of the denominator
Permutation = [tex]\frac{13!}{6!}[/tex]
Find the factorial of the values
Permutation = [tex]\frac{6,227,020,800}{720}[/tex]
Divide the numerator by the denominator
Permutation = 8, 648, 640 ways
Thus, the number of ways the program can be arranged for this segment is 8, 648, 640 ways
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PLEASE ANSWER QUICK !!!!
Mikayla is determining the actual
distance between Harrisville and Lake Town
using a map. The scale on her map reads
1 inch = 50 miles. She measures the distance
to be 4.5 inches and writes the following
proportion:
1 in. 50 mi
x mi
4.5 in.
P
Explain why her proportion is equivalent to
50 mix mi
1 in. 4.5 in."
Answer:
100 miles and 5.5 inches
the confidence interval for the population variance is
(730.72, 4569.53) is the confidence interval for the population variance σ² given that c = 0.98, s = 39 and n = 15. This can be obtained by using formula for confidence interval and using statistical table for values.
Find the confidence interval for population variance:The formula for finding the confidence interval for population variance is:
[tex]\frac{(n-1)s^{2} }{\chi^{2} _{\frac{\alpha }{2} } } < \sigma^{2} < \frac{(n-1)s^{2} }{\chi^{2} _{1-\frac{\alpha }{2} } }[/tex]
where n is the size of the sample, (n - 1) is the degrees of freedom, s is the standard deviation, α is the significance level.
Here it is given that,
confidence level = 1 - α = 0.98 significance level = α = 0.02standard deviation = 39 sample size n = 15degrees of freedom (n - 1) = 15 - 1 = 14α/2 = 0.02/2 = 0.01
1 - α/2 = 1 - 0.01 = 0.99
For (n - 1) degrees of freedom,
[tex]\chi^{2}_{\frac{\alpha}{2} }[/tex] = 29.141 , and [tex]\chi^{2}_{1-\frac{\alpha}{2} }[/tex] = 4.660 (Using statistical table)
Using the formula we get,
[tex]\frac{(n-1)s^{2} }{\chi^{2} _{\frac{\alpha }{2} } } < \sigma^{2} < \frac{(n-1)s^{2} }{\chi^{2} _{1-\frac{\alpha }{2} } }[/tex]
[tex]\frac{(14)39^{2} }{29.141} < \sigma^{2} < \frac{(14)39^{2} }{4.660}[/tex]
[tex]730.72 < \sigma^{2} < 4569.53[/tex]
The confidence interval for the population variance is (730.72, 4569.53)
Hence (730.72, 4569.53) is the confidence interval for the population variance σ² given that c = 0.98, s = 39 and n = 15.
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Translate the sentence into an equation using n to represent the unknown number. Then solve the equation for n.
a number subtracted from 46 is -21
a.
46 minus n = negative 21. n = 67
b.
46 minus 21 = n. n = 67
c.
46 minus 21 = n. n = 25
d.
46 minus n = negative 21. n = 75
Answer: a.
Step-by-step explanation:
46 is subtracted by n to get -21 so the equation is 46 - n = -21
Add n on both sides; 46 - n + n = -21 + n; 46 = -21 + n
Add 21 on both sides; 46 + 21 = -21 + n + 21; 67 = n
So n = 67
Sophia says that you can solve the problem in the example by multiplying both quantities in the ratio 60 : 36 by 1/6. Is Sophia correct? Explain.
Answer:
No, she is wrong
Step-by-step explanation:
[tex]\frac{60}{36} = \frac{5}{3}[/tex] {Simplification}
[tex]\frac{5}{3} \neq \frac{1}{6}[/tex]
Factor completely 4x^2 − 32
Answer: [tex]4(x+2\sqrt{2})(x-2\sqrt{2})[/tex]
Step-by-step explanation:
We can first take out the common factor of 4, as both 4x² and -32 are divisible by 4.
[tex]4(x^2-8)[/tex]
From here, we can assume that x²-8 is a difference of two squares even though 8 is not a perfect square.
For review, a difference of two squares [tex]a^2-b^2[/tex] can be factored into [tex](a+b)(a-b)[/tex].
[tex]4(x^2-8)\\4(x+\sqrt{8})(x-\sqrt{8})\\4(x+2\sqrt{2})(x-2\sqrt{2})[/tex]
Calculating orthogonal trajectories?
The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.
How to find the equation for the orthogonal trajectories of a given equation
In this problem we have a family of functions in implicit form, that is, a function of the form f(x, y, c) = 0. The equation of a orthogonal trajectory is always perpendicular to a particular form of a equation. First, we determine the first derivative of the given expression:
5 · x² - 2 · y² = C
10 · x - 4 · y · y' = 0
4 · y · y' = 10 · x
y' = (10 · x) / (4 · y)
y' = (5 · x) / (2 · y)
If f(x, y) = (5 · x) / (2 · y), then the differential equation for the orthogonal trajectories related to the family of functions is:
y' = - 1 / f(x, y)
y' = - (2 · y) / (5 · x)
dy / (2 · y) = - dx / (5 · x)
By indefinite integration we get the following solution to the ordinary differential equation:
(1 / 2) · ㏑ y = - (1 / 5) · ㏑ x + C
(1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C
The equation for the trajectories orthogonal to the family of functions of the form 5 · x² - 2 · y² = C is equal to (1 / 5) · ㏑ x + (1 / 2) · ㏑ y = C.
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How many three-digit multiples of 5 have three different digits and an odd tens digit?
I am getting 72. which is wrong
Answer:
68?
Step-by-step explanation:
multiples of 5 end in 5 or 0
_ _ _
last digit has 2 choices ( 5 and 0 )
1,3,5,7,9 are odd
so tens digit has 5 choices but we see that 5 is repeated so we split into more cases
when last digit is 0, tens digit has 5 choices then hundreds digit has 8 choices so 1*5*8=40
when last digit is 5, tens digit has 4 choices and hundreds digit has 7 choices (0 can't be first digit) 1*4*7=28
40+28=68
Solve:|x-3|=7 pls answer
Answer:
x=10 and x=-4
Step-by-step explanation:
When the absolute value is all by itself on one side of the equation, the next step is to remove the absolute value bars. This will create TWO separate equations.
| x - 3 | = 7
x-3 = 7 This is one of the equations. Just straight up copied the equation but without the bars.
Here is the second equation:
x - 3 = -7
It is the left side copied, without the bars, but the right side has the opposite sign.
Solve these both separately to find the answers.
x-3=7 and x-3=-7
x=10 and x=-4
A projector enlarges an image in the ratio 20:1. What will be the size of an enlargement of a picture that is 8cm by 6cm?
Answer:
160 cm × 120 cm
Step-by-step explanation:
this means 1 cm turns into 20 cm.
so, 8 cm × 6 cm becomes
8×20 = 160 cm × 6×20 = 120 cm
In a certain school 70% of the students in first-year chemistry have had algebra. If there are 290 students in first-year chemistry, how many of them have had algebra?
at the beginning of the week, a particular stock sold for 29 3/8 per share. at the end of the same week it sold for 31 2/8. what was the amount of increase per share ?
Answer:
2 1/8
Step-by-step explanation:
I counted from 29 to 31 and then saw that for 29 it was 3/8 and for 31 it was 2/8 so i knew that the answer would be 2 1/8
Consider the curve C in the Cartesian plane described in polar coordinates by given:
See picture:
a. Determine a Cartesian equation that describes curve C. Hint: first multiply (c) by r.
b Describe this curve and use this description to obtain the area inside C.
c Use (c) to set up an integral that computes the area inside C that is also within the rst quadrant.
d Evaluate this integral to determine the area.
a. Recall that in polar coordinates, we can parameterize [tex]x=r\cos(\theta)[/tex] and [tex]y=r\sin(\theta)[/tex]. So, doing as the hint suggests, we have
[tex]r = 6\cos(\theta) + 8 \sin(\theta)[/tex]
[tex]\implies r^2 = 6r\cos(\theta) + 8r\sin(\theta)[/tex]
[tex]\implies \boxed{x^2 + y^2 = 6x + 8y}[/tex]
b. By completing the square, we get
[tex]x^2 + y^2 = 6x + 8y[/tex]
[tex]x^2 - 6x + y^2 - 8y = 0[/tex]
[tex]x^2 - 6x + 9 + y^2 - 8y + 16 = 25[/tex]
[tex](x-3)^2 + (y-4)^2 = 5^2[/tex]
which is the equation of the circle centered at (3, 4) with radius 5. Thus the area bounded by [tex]C[/tex] is [tex]\pi\cdot5^2 = \boxed{25\pi}[/tex].
c. This is made easier if you can consult a plot (attached). In the first quadrant, we have [tex]0\le\theta\le\frac\pi2[/tex], while the radial coordinate [tex]r[/tex] runs uninterrupted from the origin [tex]r=0[/tex] to the circle [tex]r=6\cos(\theta)+8\sin(\theta)[/tex]. So the area is
[tex]\displaystyle \int_0^{\pi/2} \int_0^{6\cos(\theta) + 8\sin(\theta)} r\,dr\,d\theta = \boxed{\frac12 \int_0^{\pi/2} \left(6\cos(\theta) + 8\sin(\theta)\right)^2 \, d\theta}[/tex]
d. Evaluate the integral.
[tex]\displaystyle \frac12 \int_0^{\pi/2} \left(36\cos^2(\theta) + 96\sin(\theta)\cos(\theta) + 64 \sin^2(\theta)\right) \, d\theta[/tex]
Simplify the integrand with the help of the identities
[tex]\cos^2(x) + \sin^2(x) = 1[/tex]
[tex]\sin(x)\cos(x) = \dfrac12 \sin(2x)[/tex]
[tex]\sin^2(x) = \dfrac{1 - \cos(2x)}2[/tex]
[tex]\displaystyle \frac12 \int_0^{\pi/2} \left(50 + 48\sin(2\theta) - 14 \cos(2\theta)\right) \, d\theta[/tex]
The rest is easy. You should end up with
[tex]\displaystyle \frac12 \int_0^{\pi/2} \left(6\cos(\theta) + 8\sin(\theta)\right)^2 \, d\theta = \boxed{24 + \frac{25\pi}2}[/tex]
a) The Cartesian equation that described curve C is x² + y² = 6 · x + 8 · y.
b) The area inside C is A = π · 5² = 25π square units.
c) The integral that computed the area inside curve C within the first quadrant is A = (1 / 2)∫ (6 · cos θ + 8 · sin θ)² dθ, for θ ∈ [0, 0.5π].
d) The integral evaluated at the given limits is equal to an area of 20.139π square units.
How to analyze a polar equation and find its area by geometric and calculus means
In this question we find a polar equation in explicit form. a) To find the equivalent form in rectangular coordinates, we must apply the following substitutions x = r · cos θ, y = r · sin θ:
r = 6 · cos θ + 8 · sin θ
r² = 6 · r · cos θ + 8 · r · sin θ
x² + y² = 6 · x + 8 · y (1)
The Cartesian equation that described curve C is x² + y² = 6 · x + 8 · y.
b) Perhaps the equation represents a conic section, possibly a circunference. To prove this assumption, we must apply algebraic handling until standard form is obtained:
x² - 6 · x + y² - 8 · y = 0
x² - 6 · x + 9 + y² - 8 · y + 16 = 25
(x - 3)² + (y - 4)² = 5² (1b)
Which indicates a circumference centered at point (h, k) = (3, 4) and with a radius of 5 units. By the area formula for a circle we find that the area inside C is A = π · 5² = 25π square units.
c) The polar form of the area integral is presented herein:
A = ∫ ∫ r dr dθ, for r ∈ [0, r(θ)] and θ ∈ [0, 0.5π]
A = (1 / 2)∫ [r(θ)]² dθ, for θ ∈ [0, 0.5π]
A = (1 / 2)∫ (6 · cos θ + 8 · sin θ)² dθ, for θ ∈ [0, 0.5π]
The integral that computed the area inside curve C within the first quadrant is A = (1 / 2)∫ (6 · cos θ + 8 · sin θ)² dθ, for θ ∈ [0, 0.5π].
d) By algebraic handling, trigonometric formulas and integral properties:
A = 25 ∫ dθ + 24 ∫ sin 2θ dθ - 14 ∫ cos 2θ dθ, for θ ∈ [0, 0.5π]
A = 25 · θ - 12 · cos 2θ - 7 · sin 2θ, for θ ∈ [0, 0.5π]
A = 20.139π
The integral evaluated at the given limits is equal to an area of 20.139π square units.
To learn more on circumferences: https://brainly.com/question/4268218
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pLEASE answer as fast as possible REALLY URGENT
Using proportions, it is found that you would expected the white counter to be chosen 128 times.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.
From the table, the proportion of the white counter is given as follows:
p = 32/(32 + 18) = 32/50 = 0.64.
Hence, out of 200 trials, the number of trials in which the white counter is expected to be chosen is given by:
0.64 x 200 = 128 times.
More can be learned about proportions at https://brainly.com/question/24372153
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