Answer:
idek
Step-by-step explanation:
Which represents the solution set to the inequality 5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)?
x < –2.5
x > 2.5
(–2.5, ∞)
(–∞, 2.5)
The solution to the inequality is (–2.5, ∞)
How to solve the inequality?The inequality is given as:
5.1(3 + 2.2x) > –14.25 – 6(1.7x + 4)
Open the brackets
15.3 + 11.22x > –14.25 – 10.2x - 24
Collect the like terms
11.22x + 10.2x > -15.3 - 14.25- 24
Evaluate the like terms
21.42x > -53.55
Divide both sides by 21.42
x > -2.5
This can also be represented as (–2.5, ∞)
Hence, the solution to the inequality is (–2.5, ∞)
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Answer:
c
Step-by-step explanation:
on edge
The acting club two-act plays begins at 3:20 P.M. The first act is twice as long as the second act, and there is a 15-minute break between the two acts. The play ends at 4:50 P.M. How long is Act 1?
Answer:
50 minutes
Step-by-step explanation:
3:20 pm - 4:50 = 90 minutes
90 = 2a + 15 + a
90-15= 75
75= 2a + a
75 = 3a
75/3 = 25
25 x 2 = 50 minutes
A 12-ounce cup of juice contains 70 percent fruit and 30 percent water. If combined with 16 ounces of juice that contain 10 percent fruit and 90 percent water, what portion of the mixture is fruit?
Write the answer to the nearest hundredth
The portion of the mixture that is fruit to the nearest hundredth is 10.00 ounces
Percentage12 ounce juice:
Water = 30%Fruits = 70%= 70/100 × 12
= 0.7 × 12
= 8.4 ounces
Water = 30%
16 ounces;
Water = 90%Fruits = 10%= 10/100 × 16
= 0.1 × 16
= 1.6 ounces
Portion of mixture that is fruits = 8.4 ounces + 1.6 ounces
= 10.00 ounces
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A basin can hold 8542ml of water.the basin can hold 0.458l more water than a fish tank. how much water can the fish tank hold?express your answer in litres.
Answer:
8.048 liters
Step-by-step explanation:
Convert 8542 ml to liters: (8542 ml)(1 liter/1000 ml) = 8.542 liters
Now subtract 0.458 liters to find the amount that a fish tank can hold:
8.542 liters
-0.458 liters
8.048 liters is the fish tank capacity
B(x) = 0. 06x^2 - 0. 2x^3, find the dosage at which the resulting blood presure is maximized
The dosage at which the resulting blood pressure is maximized is x at 0.2 and the maximum dosage is 0.0008.
In the question,
The function is [tex]B(x) = 0. 06x^2 - 0. 2x^3[/tex]
To find the maximum or minimum, take the derivative and set it equal to zero.
⇒ [tex]B'(x) = 2(0. 06)x - 3(0. 2)x^{2}[/tex]
Setting it equal to zero, we get
⇒ [tex]0.12x - 0.6x^{2}=0[/tex]
⇒ 0.6x (0.2-x) = 0
⇒ x = 0 or x = 0.2
Now substitute x = 0.2 in B(x), we get
⇒ [tex]B(0.2) = 0. 06(0.2)^{2} - 0. 2(0.2)^{3}[/tex]
⇒ B(0.2) = 0.0024 - 0.0016
⇒ B(0.2) = 0.0008
To know B(0.2) is maximum, let us find the values for x = 1 and x = 0.01.
For x = 1,
⇒ [tex]B(1) = 0. 06(1)^{2} - 0. 2(1)^{3}[/tex]
⇒ B(1) = 0.06-0.2
⇒ B(1) = -1.04
For x = 0.01,
⇒ [tex]B(0.01) = 0. 06(0.01)^{2} - 0. 2(0.01)^{3}[/tex]
⇒ B(0.01) = 0.000006 - 0.0000002
⇒ B(0.01) = 0.0000058
Thus, x = 0.2 is the maximum.
Hence we can conclude that the dosage at which the resulting blood pressure is maximized is x at 0.2 and the maximum dosage is 0.0008.
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A farmer uses a lot of fertilizer to grow his crops. The farmer's manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B's fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 33 pounds per batch and fertilizer from distributor B contained 25 pounds per batch. Suppose the population standard deviation for distributor A and distributor B is three pounds per batch and four pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let µ1 and µ2 represent the average amount of nitrogen per batch for fertilizer A and B, respectively. Which of the following is the correct value of the test statistic?
Based on the calculations, the correct value of the test statistic is equal to 3.2.
How to calculate value of the test statistic?For samples A and B, the hypothesis is given by:
H₀: μ₁ ≤ μ₂
H₁: μ₁ > μ₂
Since both samples have a normal distribution, we would use a pooled z-test to determine the value of the test statistic:
[tex]z = \frac{\bar{x_1} - \bar{x_2} -(\mu_1 - \mu_2) }{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_1^2}{n_1}} }[/tex]
Substituting the given parameters into the formula, we have;
[tex]z = \frac{33 - 25 -(0) }{\sqrt{\frac{3^2}{4} + \frac{4^2}{4}} }\\\\z = \frac{8 }{\sqrt{\frac{9}{4} + \frac{16}{4}} }\\\\z = \frac{8 }{\sqrt{\frac{25}{4} } }\\\\z = \frac{8 }{\frac{5}{2} }}\\\\z = 8 \times \frac{2}{5}[/tex]
z = 3.2.
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Determine the unknown length or angle measurement. Round each answer to the nearest whole number
Answer:
Θ ≈ 37° , x ≈ 13 cm
Step-by-step explanation:
(a)
using the cosine ratio in the right triangle
cosΘ = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BC}{AC}[/tex] = [tex]\frac{8}{10}[/tex] , then
Θ = [tex]cos^{-1}[/tex] ([tex]\frac{8}{10}[/tex] ) ≈ 37° ( to the nearest whole number )
(b)
using the sine ratio in the right triangle and the exact value
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , then
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{x}{15}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2x = 15[tex]\sqrt{3}[/tex] ( divide both sides by 2 )
x = 7.5[tex]\sqrt{3}[/tex] ≈ 13 cm ( to the nearest whole number )
Mark each statement as true or false. Suppose A is an n n matrix. a. If an n × n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable. False b. If A is diagonalizable, then A is also diagonalizable False c. If there is a basis of R n consisting of eigenvectors of A, then A is diagonalizable. True d. A is diagonalizable if and only if A has n eigenvalues, counting multiplicity. False e. If A is diagonalizable, then A is invertible. False
The correct option for the matrix will be:
FalseTrueTrueFalseFalseHow to explain the matrix?
a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
FalseIt could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.
b) If A is diagonalizable the A2 is diagonalizable
TrueIf A is diagonalizable then there exists an invertible matrix
c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.
Trued) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.
Falsee) If A is diagonalizable, then A is invertible.
FalseIt’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.
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100 families booked a holiday in July or in August, at travel agents.
Some of the families booked to go to France.
Some booked to go to Spain.
The rest of the families booked a holiday to Portugal.
59 families booked to go on holiday in August. 19 of the 35 families going to France booked to go in July.
30 families booked to go to Portugal.
20 families booked to go to Spain in August. How many families booked to go to Portugal in July?
Answer:
7
Step-by-step explanation:
A 2-way table can be useful for recording the given information and for finding the missing numbers.
SetupThe attached table shows the numbers of families booking in July and August, and also counted by destination. Totals are at the right and bottom, and the grand total is the number 100 at the lower right.
Underlined numbers are those in the problem statement. The remaining numbers are computed so as to make the totals be correct.
SolutionOverall, 30+35 = 65 went to Portugal or France, so 100-65 = 35 went to Spain. Of those, 20 booked in August, so 15 booked to Spain in July.
59 booked in August, so a total of 41 booked in July.
Now we know 19 booked for France and 15 booked for Spain in July, so the remaining 7 booked for Portugal in July.
Ralph Warren purchased 27 shares of stock at 16 3/8 per share. He paid a $27.50 brokerage fee. He later sold all 27 shares at 17 5/8 and paid a $28.75 brokerage fee. (36) What was his total cost for the stock including his brokerage fee? (37) What did he receive from the sale of the stock after he paid the brokerage fee? (38) Did he have a capital gain or loss? (39) How much was the gain or loss? (40) What was the net change from 16 3/8 to 17 5/8?
Answer:
Ralph Warren purchased 27 shares of stock at 16 3/8 per share. He paid a $27.50 brokerage fee. He later sold all 27 shares at 17 5/8 and paid a $28.75 brokerage fee. (36) What was his total cost for the stock including his brokerage fee? (37) What did he receive from the sale of the stock after he paid the brokerage fee? (38) Did he have a capital gain or loss? (39) How much was the gain or loss? (40) What was the net change from 16 3/8 to 17 5/8?
Step-by-step explanation:
Additionally, mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. In your initial post for this discussion, address both of the following:What are differences or similarities between everyday logic and mathematical logic?How can the study of mathematical logic help you in your everyday life?
Mathematical logic is important as it's a way to learn new experience through continuous self assessment.
How to illustrate the information?Logic is important as it enables us to form sound judgements and beliefs.
The study of logic can help as it can help us to understand disagreement and ambiguity.
It also helps us in making a reasonable emotional life.
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There are Z fish in a aquarium. 1/4 of the fish are angelfish. How may are angelfish?
Step-by-step explanation:
since there are z fishes
the number of angelfish = 1 × z
4
= 1 z
4
x-15/3 = 18x. How to solve this question? I will mark the first answerer as brainliest.
How many palindromes greater than 10000 and less than 100000 are multiples of 18?
Let [tex]n=abcba[/tex] be such a number. If 18 divides [tex]n[/tex], then both 2 and 9 divide [tex]n[/tex].
To be divisible by 2, we must have [tex]a\in\{2,4,6,8\}[/tex]. Meanwhile we can have [tex](b,c)\in\{0,1,2,\ldots,9\}^2[/tex].
To be divisible by 9, we the sum of the digits of [tex]n[/tex] must itself be divisible by 9, or
[tex]2a+2b+c=9k[/tex]
for some integer [tex]k[/tex].
The largest value of [tex]2a+2b+c[/tex] is 2•8 + 2•9 + 9 = 43, so we must have [tex]k\in\{1,2,3,4\}[/tex].
I'm not sure what the best way to get the final count may be, but there are 44 such numbers. It's rather tedious to do by hand.
• If [tex]k=1[/tex], then [tex]2a+2b+c=9[/tex], and we can do this in 4 ways.
For example,
2•2 + 2•0 + 5 = 9 [tex](n = 20502)[/tex]
• If [tex]k=2[/tex], then [tex]2a+2b+c=18[/tex] and can be done in 16 ways.
2•2 + 2•3 + 8 = 18 [tex](n = 23832)[/tex]
• If [tex]k=3[/tex], then [tex]2a+2b+c=27[/tex] and can be done in 18 ways.
2•2 + 2•7 + 9 = 27 [tex](n = 27972)[/tex]
• If [tex]k=4[/tex], then [tex]2a+2b+c=36[/tex] and can be done in 6 ways.
2•6 + 2•8 + 8 = 36 [tex](n = 68886)[/tex]
a group of young women decided to raise 480000 to start a business after some time 4 women pulled outand they had to pay additional 20000 . determine the original number of women
Answer:
96 is the number of original women investors.
Step-by-step explanation:
Let X be the number of young women in the initial group. They raised 480000, so the average payment per person was (480000/X).
When 4 pull out, the new average is (480000/(X-4)). We are told that this new average required the remaining women (X-4) to add another 20000.
The four women therefore had contributed 20000 in total, making their average 20000/4 = 5000 each.
This would have been the same amount contributed by all X women. Thus, we can set the average (480000/X) equal to 5000
(480000/X) = 5000
(480000) = 5000X
(480000)/5000 = X
X = 96
The original number of women was 96.
===
Check
(96 Women)(5000/Woman) = 480000 CHECKS
(4 Women pull out)*(5000) = 20000 that needs to be added to stay at 480000. CHECKS
Find the discriminant of the quadratic equation x2 6x 14 = 0 and use it to determine the number and types of solutions. b2 − 4ac −20; two nonreal solutions −20; one real solution 92; two real solutions 92; one real solution
The discriminant of the quadratic equation is -20 and there are two non real solutions.
The discriminant of a quadratic equation uses the equation:
[tex]b^{2} -4ac[/tex]
Where the value of this calculation can tell you what solutions there are, plug known values in:
[tex]x^{2} +6x+14[/tex]
a=1
b=6
c=14
[tex]b^{2} -4ac[/tex]
which is equal to -20 on putting the values of a,b and c in the equation
As -20 < 0, this means that there is not a real solution, resulting in the first option being correct.
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In a bag of 10 marbles, there are 4 blue, 3 red, 2 green,
and 1 yellow. What is the probability that you draw one
marble that is red, replace it, and draw another marble that
is yellow?
a. 2
-
5
b. 3
100
C. 1
30
O d. 3
10
[tex]P(R \: then \: Y) = \frac{3}{10} \times \frac{1}{10} = \frac{3}{100} [/tex]
[tex]note \: that = \\ order \: matters \: (no \: factorial) \\ replacement \: (equal \: sample \: space)[/tex]
Option BAnswer:
c. 3/100
Step-by-step explanation:
there are 10 marbles
First draw (there are 3 red marbles)
[tex]P=\frac{3}{10}[/tex]
Second draw (there is one yellow marble)
[tex]P=\frac{1}{10}[/tex]
probability of the event:
[tex]P=(\frac{3}{10} )(\frac{1}{10} )=\frac{3}{100}[/tex]
Hope this helps
Which numbers are integers? Check all that apply.
4
Negative 1 and one-third
-10
2.5
-4
0.ModifyingAbove 13 with Bar
YOOOO QUICK I REALLY NEED THIS PLEASEEE
The numbers : 4,-10 and -4 are integers.
A number without a decimal or fractional element is known as an integer, which can be both positive and negative, including zero.
The Latin word "integer" signifies "whole" or "intact." Thus, fractions and decimals are not included in integers.
All whole numbers and negative numbers are considered integers. This means that if we combine negative numbers with whole numbers, a collection of integers results.
"Negative 1 and one-third" includes a fractional part, so it is not an integer.
"2.5" and "0.ModifyingAbove 13 with Bar" contain a decimal, so it is also not an integer.
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1. Reflect Rectangle ABCD across the y-axis, then
translate it using the following rule:
(x,y) → (x-4, y - 3).
A B
D
с
-4 -2
4
2
ТУ
O
-2-
-4-
2
4
X
√x
A' (
B'(
C'(
D'(
A"(
B"(
C"(
D"(
)
)
)
)
PLS HELP ME WITH THIS
Answer:
See attached image
Step-by-step explanation:
Select the correct answer from the drop-down menu.
The expression
is not equivalent to (1 - sin²(x)) tan(-x).
J
e
C
✔
The expression that is not equivalent to (1 - sin²(x)) tan(-x) is D. (cos²x - 1)(cot -x).
How to illustrate the expression?It should be noted that (cos²x - 1)(cot -x). is not equivalent to the given expression.
This is illustrated as:
(cos²x - 1)(cot -x)
= (-sin²x) × (-cot x)
= sin²x × cosx/sin x = sinxcosx
In conclusion, the correct option is D.
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The expression _______ is not equivalent to (1 − sin2(x)) tan(-x).
a. (1 - cos^2(x)) cot(-x)
b. (cos^2(x) - 1) cot(x)
c. (sin^(x) - 1) tan(x)
d. (cos^2(x) - 1) cot(-x)
PLEASE HELP!!!! I don’t understand this topic please can someone help explain how to work out the attached question on similar area. Will mark brainliest!
Step-by-step explanation:
Look, you're taking these kinds of questions too serious and that's why you believe they're hard, now pay close attention:
the areas are defined in cm² but the radius is defined in cm, so you need find the positive root of cm².
[tex] \sqrt{ \frac{5500 {cm}^{2} }{220 {cm}^{2} } } = \frac{r}{5cm} = = > \\ 5 = \frac{r}{5} = = = > r = 25[/tex]
and the question wants the base area:
[tex]s = \pi {r}^{2} = = = > s = 3.14 \times {(25cm)}^{2} = = > 1963.5 {cm}^{2} [/tex]
Triangle ΔABC is reflected across line n to create ΔA'B'C'
What is the measure of ∠C?
Answer:
54 degrees
Step-by-step explanation:
The actual angles of the triangle are not changing since this is just a reflection. The angles inside a triangle all add up to 180, so we can do 180-59-67 to get 54.
PLEASE HELP!!! I'LL MARK AS BRAINLIEST TO THE FIRST PERSON THAT CAN ANSWER!!!
When two exponents with the same base are multiplied together, this reflects the _______ of powers property.
solve the equation below by factorising.
2x^2-8x=0
Hello,
2x² - 8x = 0
2x × x - 2x × 4 = 0
2x(x - 4) = 0
2x = 0 or x - 4 = 0
x = 0 or x = 4
A rectangular closet has a perimeter of 18 feet and an area of 20 square feet. What are the dimensions of the closet.
Answer:
The width can either be 4 or 5 feet.
Step-by-step explanation:
Assuming the length is x and the width is y.2x+2y=18xy=202y=18−2xy=9−xx(9−x)=209x−x2=200=x2−9x+200=(x−5)(x−4)x=5and4 The width can either be 4 or 5 feet
A basketball player has a 0.689 probability of making a free throw. If the player shoots 18 free throws, what is the probability that she makes no more than 11 of them
It is determined while using the binomial distribution that there is still a 1.145=114.5% chance that she produces no more than 11 of them.
Calculating the probabilityThere are just two possible results for each throw. Either she succeeds or she fails. The binomial probability distribution is employed to answer this issue since the probability of completing a shot is regardless of all other throws.
Binomial probability distribution-
[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]
[tex]C_{n,x} = n!/x! (n-x)![/tex]
where,
the no. of success= x
the no. of trials = n
the probability of a success on one trial = p
The probability of throwing not more than 11 will be:
P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)
Where,
[tex]P(X=x) = C_{n,x} .p^{x}(1-p)^{n-x}[/tex]
[tex]P(X=0) = C_{18,0} .(0.689)^{0}(0.311)^{18}[/tex]≈0
[tex]P(X=1) = C_{18,1} .(0.689)^{1}(0.311)^{17}[/tex]≈0
[tex]P(X=2) = C_{18,2} .(0.689)^{2}(0.311)^{16}[/tex]≈0
[tex]P(X=3) = C_{18,3} .(0.689)^{3}(0.311)^{15}[/tex]≈0
[tex]P(X=4) = C_{18,4} .(0.689)^{4}(0.311)^{14}[/tex]≈0
[tex]P(X=5) = C_{18,5} .(0.689)^{5}(0.311)^{13}[/tex]= 0.0003
[tex]P(X=6) = C_{18,4} .(0.689)^{6}(0.311)^{12}[/tex]=0.0016
[tex]P(X=7) = C_{18,7} .(0.689)^{7}(0.311)^{11}[/tex]=0.0062
[tex]P(X=8) = C_{18,8} .(0.689)^{8}(0.311)^{10}[/tex]=0.0188
[tex]P(X=9) = C_{18,9} .(0.689)^{9}(0.311)^{9}[/tex]=0.0463
[tex]P(X=10) = C_{18,10} .(0.689)^{10}(0.311)^{8}[/tex]=0.9232
[tex]P(X=11) = C_{18,11} .(0.689)^{11}(0.311)^{7}[/tex]=0.1488
So,
P(X<11) = P(X=0) + P(X=1) +P(X=2) + P(X=3) +P(X=4) +P(X=5)+P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)
=0+0+0+0+0+0.0003+0.0016+0.0062+0.0188+0.0463+0.9232+0.1488 =1.145
Therefore, she makes 1.145=114.5% probability, no more than 11 of them.
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Evaluate the interval (Calculus 2)
Answer:
[tex]2 \tan (6x)+2 \sec (6x)+\text{C}[/tex]
Step-by-step explanation:
Fundamental Theorem of Calculus
[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.
Given indefinite integral:
[tex]\displaystyle \int \dfrac{12}{1-\sin (6x)}\:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int a\:\text{f}(x)\:\text{d}x=a \int \text{f}(x) \:\text{d}x$\end{minipage}}[/tex]
If the terms are multiplied by constants, take them outside the integral:
[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)}\:\:\text{d}x[/tex]
Multiply by the conjugate of 1 - sin(6x) :
[tex]\implies 12\displaystyle \int \dfrac{1}{1-\sin (6x)} \cdot \dfrac{1+\sin(6x)}{1+\sin(6x)}\:\:\text{d}x[/tex]
[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{1-\sin^2(6x)} \:\:\text{d}x[/tex]
[tex]\textsf{Use the identity} \quad \sin^2 x+ \cos^2 x=1:[/tex]
[tex]\implies \sin^2 (6x) + \cos^2 (6x)=1[/tex]
[tex]\implies \cos^2 (6x)=1- \sin^2 (6x)[/tex]
[tex]\implies 12\displaystyle \int \dfrac{1+\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]
Expand:
[tex]\implies 12\displaystyle \int \dfrac{1}{\cos^2(6x)}+\dfrac{\sin(6x)}{\cos^2(6x)} \:\:\text{d}x[/tex]
[tex]\textsf{Use the identities }\:\: \sec \theta=\dfrac{1}{\cos \theta} \textsf{ and } \tan\theta=\dfrac{\sin \theta}{\cos \theta}:[/tex]
[tex]\implies 12\displaystyle \int \sec^2(6x)+\dfrac{\tan(6x)}{\cos(6x)} \:\:\text{d}x[/tex]
[tex]\implies 12\displaystyle \int \sec^2(6x)+\tan(6x)\sec(6x) \:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating $\sec^2 kx$}\\\\$\displaystyle \int \sec^2 kx\:\text{d}x=\dfrac{1}{k} \tan kx\:\:(+\text{C})$\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{6 cm}\underline{Integrating $ \sec kx \tan kx$}\\\\$\displaystyle \int \sec kx \tan kx\:\text{d}x= \dfrac{1}{k}\sec kx\:\:(+\text{C})$\end{minipage}}[/tex]
[tex]\implies 12 \left[\dfrac{1}{6} \tan (6x)+\dfrac{1}{6} \sec (6x) \right]+\text{C}[/tex]
Simplify:
[tex]\implies \dfrac{12}{6} \tan (6x)+\dfrac{12}{6} \sec (6x)+\text{C}[/tex]
[tex]\implies 2 \tan (6x)+2 \sec (6x)+\text{C}[/tex]
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Substitute [tex]y=6x[/tex] and [tex]dy=6\,dx[/tex] to transform the integral to
[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = 2 \int \frac{dy}{1 - \sin(y)}[/tex]
Now substitute [tex]t=\tan\left(\frac y2\right)[/tex] and [tex]dt=\frac12 \sec^2\left(\frac y2\right) \, dy[/tex] to transform this to
[tex]\displaystyle 2 \int \frac{dy}{1 - \sin(y)} = 2 \int \frac1{1-\frac{2t}{1+t^2}}\cdot\frac{2\,dt}{1+t^2} = 4 \int \frac{dt}{(t-1)^2}[/tex]
Finally, substitute [tex]s = t-1[/tex] and [tex]ds=dt[/tex] to get
[tex]\displaystyle 4 \int \frac{dt}{(t-1)^2} = 4 \int \frac{ds}{s^2} = -\dfrac4s + C[/tex]
Now recover the antiderivative in terms of [tex]x[/tex].
[tex]\displaystyle \int \frac{12}{1-\sin(6x)} \, dx = -\frac4s + C \\\\ ~~~~~~~~ = -\frac4{t-1} + C \\\\ ~~~~~~~~ = -\frac4{\tan\left(\frac y2\right) - 1} + C \\\\ ~~~~~~~~ = \boxed{-\frac4{\tan(3x) - 1} + C}[/tex]
On a certain math exam, $10\%$ of the students got 70 points, $25\%$ got 80 points, $20\%$ got 85 points, $15\%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam
Solve the system of equations 4x+5y=-1 and -5x-8y=10 by combining the equations.
The solution of the equation are as follows:
x = 6 and y = -5
How to solve the system of equation?4x + 5y = -1
-5x - 8y = 10
Therefore,
20x + 25y = -5
-20x - 32y = 40
-7y = 35
y = -5
Hence,
4x + 5(-5) = -1
4x - 25 = -1
4x = -1 + 25
4x = 24
x = 24 / 4
x = 6
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Explain how you can find the constant of proportionality from a graph representing a proportional relationship when it shows a point with an x-value of 1 and if it doesn’t show an x-value of 1.
For a proportional relationship, the constant is found dividing all values of y by each respective value of x.
What is a proportional relationship?A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:
y = kx
In which k is the constant of proportionality.
The constant can be represented as follows:
[tex]k = \frac{y}{x}[/tex]
Hence the constant is found dividing all values of y by each respective value of x.
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