(a) Recall that
[tex]\dfrac{\mathrm d[\sin^{-1}x]}{\mathrm dx}=\dfrac1{\sqrt{1-x^2}}[/tex]
[tex]\dfrac{\mathrm d[\ln x]}{\mathrm dx}=\dfrac1x[/tex]
[tex]\dfrac{\mathrm d[\sqrt x]}{\mathrm dx}=\dfrac1{2\sqrt x}[/tex]
Then by the chain rule,
[tex]y=\sin^{-1}(\sqrt x)+\ln(\sqrt x)[/tex]
[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm d[\sin^{-1}(\sqrt x)+\ln(\sqrt x)]}{\mathrm dx}[/tex]
[tex]=\dfrac1{\sqrt{1-(\sqrt x)^2}}\dfrac{\mathrm d[\sqrt x]}{\mathrm dx}+\dfrac1{\sqrt x}\dfrac{\mathrm d[\sqrt x]}{\mathrm dx}[/tex]
[tex]=\dfrac1{2\sqrt x\sqrt{1-x}}+\dfrac1{2(\sqrt x)^2}[/tex]
[tex]=\dfrac1{2\sqrt{x-x^2}}+\dfrac1{2x}[/tex]
[tex]=\boxed{\dfrac{x+\sqrt{x-x^2}}{2x\sqrt{x-x^2}}}[/tex]
(b) I'll assume [tex]\log x[/tex] refers to the logarithm of base [tex]e[/tex]. Recall that
[tex]\dfrac{\mathrm d[e^x]}{\mathrm dx}=e^x[/tex]
[tex]2^x=e^{\log2\,x}\implies\dfrac{\mathrm d[2^x]}{\mathrm dx}=e^{\log2\,x}\dfrac{\mathrm d[\log2\,x]}{\mathrm dx}=\log2\cdot2^x[/tex]
Then using the chain rule and implicit differentiation, we get
[tex]e^y=2^x\log x-e^{2x}[/tex]
[tex]\implies e^y\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm d[2^x]}{\mathrm dx}\log x+2^x\dfrac{\mathrm d[\log x]}{\mathrm dx}-e^{2x}\dfrac{\mathrm d[2x]}{\mathrm dx}[/tex]
[tex]e^y\dfrac{\mathrm dy}{\mathrm dx}=\log2\cdot2^x\log x+\dfrac{2^x}x-2e^{2x}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\log2\cdot2^x\log x+\frac{2^x}x-2e^{2x}}{e^y}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\log2\cdot2^x\log x+\frac{2^x}x-2e^{2x}}{2^x\log x-e^{2x}}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\boxed{\dfrac{\log2\cdot2^xx\log x+2^x-2xe^{2x}}{x(2^x\log x-e^{2x})}}[/tex]
If instead [tex]\log x=\log_{10}x[/tex] denotes the base 10 logarithm, you need only make the following adjustment:
[tex]\log x=\dfrac{\ln x}{\ln 10}\implies\dfrac{\mathrm d[\log_{10}x]}{\mathrm dx}=\dfrac1{\ln10\,x}[/tex]
where [tex]\ln x[/tex] is logarithm of base [tex]e[/tex].
In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 53.5 inches, and standard deviation of 7.3 inches.
A) What is the probability that a randomly chosen child has a height of less than 52.85 inches?
B) What is the probability that a randomly chosen child has a height of more than 46.8 inches?
Answer:
A) 0.46452
B) 0.82064
Step-by-step explanation:
We solve for question A and B using z score formula
z = (x - μ)/σ,
where x is the raw score
μ is the population mean
σ is the population standard deviation.
A) What is the probability that a randomly chosen child has a height of less than 52.85 inches?
x = 52.85 inches, μ = 53.5 inches, σ = 7.3 inches
z = (x - μ)/σ
= 52.85 - 53.5 / 7.3
= -0.08904
Using the z table to find the probability of the z score above.
P(x<52.85) = 0.46452
Therefore, the probability that a randomly chosen child has a height of less than 52.85 inches is 0.46452
B) What is the probability that a randomly chosen child has a height of more than 46.8 inches?
x = 46.8 inches, μ = 53.5 inches, σ = 7.3 inches
z = (x - μ)/σ
= 46.8 - 53.5 / 7.3
= -0.91781
Using the z table to find the probability of the z score above.
P(x<46.8) = 0.17936
P(x>46.8) = 1 - P(x<46.8)
= 1 - 0.17936
= 0.82064
Therefore, the probability that a randomly chosen child has a height of more than 46.8 inches is 0.82064
For a parade, attendance was rounded as 10,000 to the nearest hundred. What was the greatest and least number of people that could have attended the parade? Explain your answer.
Answer:
Greatest = 10,049
Least = 9,950
Step-by-step explanation:
Given that:
attendance was rounded as 10,000 to the nearest hundred
The greatest number of people that could have attended the parade is 10,049, this is because, the digit in the hundred place is 0, hence, since the next digit after 0 is less than 5, all subsequent digits are rounded down to 0 = 10,000.
If the number is goes 1 above 10,049 to 10,050, the rounding to the nearest hundred gives 10,100
The least number of people that could have attended should be 9,950; the digit in the hundred place is 9, since the digit after 9 is up to 5, then it is rounded to 100 and added to 9,900 to give 10,000.
If the number goes 1 less than 9950 to 9949, then rounding to the nearest hundred gives 9900.
Given the following definitions:
U = {1, 2, 3, 4, 5, 6, 7}
A = {1, 2, 4, 5)
B = {1,3,5,7)
How many elements are in A'n B?
Step-by-step explanation:
First,
(A intersect B) = {1,2,4,5} intersect {1,3,5,7}
= {1,5}
Now,
A'n B = (A) - (A intersect B)
= {1,2,4,5} - {1,5}
= {2,4}
A map of a park says its scale is 1 to 100 What do you think that means? Type your answer here
1cm:100m
Step-by-step explanation:
It means for every 1 cm in the map 100 m will be represented correspondingly..
Hence.. 2 cm:200m and so on..
Write a decimal expansion for 2/9. Chose the correct decimal notation.
Answer:
The answer is
2/9=0.222222222.
How much will it cost to print a circular sign with a radius of 1.5 feet if the printing company charges $29 per square foot? Use A = πr2, where π is approximately 3.14. (Round your answer to the nearest cent.)
Answer:
$204.89
Step-by-step explanation:
Step 1
We have to first find the area of the circular sign.
Since the sign is circular in shape, we use the Area of Circle.
Area of a Circle = πr²
r = 1.5 feet
π = 3.14
Area of a Circle = 3.14 × 1.5²
= 7.065ft² or 7.065 square feet.
Step 2
We find out how much it cost to print the circular sign.
We are told in the question that
The printing company charges $29 per square foot
Hence,
1 square foot = $29
7.065 square feet =
Cross multiply
= 7.065 × $29
=$ 204.885
Approximately to the nearest cent, $204.89
Therefore, it will cost $204.89 to print the circular sign
If x^2 = 196, what is the sum of the possible values of (x −1)^2?
Answer:
Step-by-step explanation:
x²=196
x=±14
when x=14
(x-1)²=(14-1)²=13²=169
when x=-14
(x-1)²=(-14-1)²=(-15)²=225
sum=169+225=394
How to 2-(-5)+1 simplified
Answer:
8
Step-by-step explanation:
We can simplify this expression by solving it.
[tex]2-(-5)+1[/tex]
Subtracting a negative is the same as adding a positive:
[tex]2+5+1[/tex]
And addition here shows that [tex]2+5+1=8[/tex].
Hope this helped!
Answer:
8Step-by-step explanation:
[tex]2-\left(-5\right)+1\\\\Follow\:the\:PEMDAS\:order\:of\:operations\\\\\mathrm{Apply\:rule\:}-\left(-a\right)\:=\:+a\\\\-\left(-5\right)=+5\\\\=2+5+1\\\\2+5=7\\\\= 7+1 \\\\=8[/tex]
A had $360 more than B. When A spent $144 and B spent half of his money, A had 6 times as much money as B, Find the ratio of the amount of money A had at first to that of B. Find the ratio of the amount of money A had at first to that of B. Give your answer in its simplest form.
Answer:
13:3
Step-by-step explanation:
A=x B=y
first; x= 360+y
later;x= 216+y
A:B
216+y: 1/2y = 6: 1
216+y/y÷2 =6/1
216+y = 3y
y = 108
first; x =360+y
360 +108
468
A:B = 468:108
13:3
Plz write all the steps
Answer:
Step-by-step explanation:
See attachment
A diver descended 20 feet in 5 seconds. What is the average rate per second that he traveled? Explain how you can use integer tiles to solve the problem and find the answer. Pls help
Answer:
The diver is moving down, so he moved −20 feet. Start with 20 negative tiles and separate them into 5 equal groups. There are 4 negative tiles in each group, so he traveled at a rate of −4 feet per second.
Step-by-step explanation:
Answer:
4 feet per second
Step-by-step explanation:
We can create a proportion:
[tex]\frac{20}{5} = \frac{x}{1}[/tex]
Cross multiply:
[tex]20\cdot1=20\\\\20\div5=4[/tex]
So he's travelling at a rate of 4 feet per second.
Hope this helped!
The graphs of linear functions f and g are shown. Enter the solution to the equation f(x)=g(x).
The two diagonal lines cross at (-3,5). We only need to worry about the x coordinate here, so x = -3 is what we're after. If x = -3, then f(x) and g(x) have the same y output meaning f(x) = g(x). That y output is y = 5.
compare your answers from parts A and B. What connections can you make between the size and number of squares in part A and the prime factors in part B?
Answer:
In part A, 6 squares made up of 4 units represented an area of 24.
In part B, the prime factorization of 24 gave 24 = 2 ∙ 2 · 2 · 3, which is equal to 4 · 6.
The factor that is a perfect square, in this case 4, determines the size of squares needed for the visual model, and other factor, 6, determines the number of squares of that size.
Step-by-step explanation:
What is the remainder when f(x) = x^2 + 14x − 8 is divided by (x − 5)? 103 88 87 72
Answer:
The answer is 87Step-by-step explanation:
f(x) = x² + 14x - 5
To find the reminder when f(x) is divided by x - 5 , substitute the value of x into the above formula
That's
x - 5 = 0
x = 5
So we have
f(5) = 5² + 14(5) - 8
f(5) = 25 + 70 - 8
f(5) = 95 - 8
We have the final answer as
87Hope this helps you
Answer: 87
Step-by-step explanation: took the test hope this helps :)
Explain how to use the distributive property to find an expression that is equivalent to 20+10
Answer: 2(10+5)
Step-by-step explanation: half of 20 is 10. half of 10 is five and i put them both in parentheses. outside the parenthesis i placed 2 so it is multiplied to the inner set of numbers.
Think I might need little assistance
Answer:
x = 90
Step-by-step explanation:
We use the Definition of Corresponding Angles to help us solve for x:
∠1 = ∠4
Step 1: Write out equation
3x - 160 = x + 20
Step 2: Solve for x
2x - 160 = 20
2x = 180
x = 90
19. A game is played by rolling a group of at least 3 dice. The highest two numeric results are removed
from the table. If any 6s remain on the table the player wins. If only 3 dice are rolled then the
probability of winning is 1/216. As the number of dice used increases without bound what does the
probability of a winning roll approach?
Answer: Approaches to 1.
Step-by-step explanation:
If there are only 3 dice used, then the only chance that the player has to win is when the 3 dice have the same outcome, 6.
The probability will be:
p = (1/6)^3 = 1/216.
Now, if we add one more dice, we still need at least 3 sixes to win, but the other dice can have any other value. so now the probabilities are:
dice 1---- outcome = 6, prob = 1/6.
dice 2---- outcome = 6, prob = 1/6.
dice 3---- outcome = 6, prob = 1/6.
dice 4---- outcome = any number, prob = 1.
The probability for this arrangement is still:
p = 1/216.
But now we have permutations!.
The dice that can be any number has 4 possible positions, so the actual probability will be:
P = 4*p = 4/216.
Now remember that if we have N elements, the total number of combinations of K elements ( N ≥ K) is:
[tex]C(N, K) = \frac{N!}{(N - K)!K!}[/tex]
if we add other dice, then we will have 5 dices, and 2 of them that can not be 6 that can take any position, then the number of combinations will be:
[tex]C(5, 2) = \frac{5!}{(5 - 2)!2!} = \frac{5*4}{2} = 10[/tex]
Then the probability will be:
P = 10*p = 10/216.
So we can start to see a pattern here, if we have N dices, we still only need 3 of them to be strictly 6, then we have (N - 3) dices that can be any number.
Then the probabilty of winning if you have N dices is:
P = C(N, N - 3)*p = C(N, N - 3)*(1/216)
Then as N increases, we will see that the probability tends to 1, (it actually grows larger than that, but remember that the probability is a number between 0 and 1, so the maximum is 1)
Why? well... if you roll a lot of dice, suppose 1000 of them, is really likely to have at least 3 sixes in there, so as the number of dice increases, also does the probability.
Which measure of central tendency is more representative of the typical observation if the graph of the data is skewed to the right?
Answer:
Median
Step-by-step explanation:
The measure of central tendency is used to represent an entire group of data with a common single number. There are three types of measure of central tendency, the mean , the median and the mode. Our focus here is on the median because in any skewed dataset, The median is the best form for a skewed type of distribution. The median is the middle number in an ordered set of data.
In a skewed type of distribution, the mode is the highest point of the distribution. In the measure of central tendency, the mean is mostly influenced by the outliers and it is pulled towards to the tail region of the distribution.
Which is an equation of a line that has a slope of -1/3 and passes through the point (-5, 2)?
Answer:
y= -⅓x +⅓
Step-by-step explanation:
The equation of a line is usually written in the form of y=mx+c, where m is the gradient and c is the y-intercept.
This form is also known as the point-slope form.
Since the slope is given to be -⅓, m= -⅓.
y= -⅓x +c
To find the value of c, substitute a pair of coordinates into the equation.
When x= -5, y=2,
2= -⅓(-5) +c
[tex]2 = \frac{5}{3} + c \\ c = 2 - \frac{5}{3} \\ c = \frac{1}{3} [/tex]
Thus, the equation of the line is y= -⅓x +⅓.
which statement is true regarding the graphed functions?
This is because the red and blue lines cross at (0,-2). We don't really need to worry about the y coordinate here. For this problem, all we care about is the x coordinate, which is x = 0.
When x = 0, the outputs of each function f(x) and g(x) are both the same value. So that's why we can say f(0) = g(0). It's the same as saying f(x) = g(x) has the solution x = 0.
The second side of a triangular deck is 6 feet longer than the shortest side, and the third side is 6 feet shorter than twice the length of the shortest side. If the perimeter of the deck is 72 feet, what are the length of the three sides?
Answer:
let the shortest side be 'x' ft
length of second side = x + 6 ft
length of third side = 2(x+6) - 6 ft
perimeter of triangle = 72 feet
shortest side + second side + third side = 72 ft
x + x + 6 + 2(x+6) - 6 = 72
4x + 12 = 72
x = 15
length of shortest side = 15 ft
length of second side = 15 + 6 = 21 ft
length of third side = 2(15 + 6) - 6 = 36 ft
Which of the following steps would not be necessary when using the square root property to solve a quadratic equation?
A.After applying the square root property, solve the resulting equations.
B.The square root property may be applied only if the constant is positive
C.When taking the square root of both sides use on the square root of the constant.
D.Isolate the quantity being squared
Answer:
B- The square root property may be applied only if the constant is positive
Max’s car holds 18 gallons of gasoline. Driving on the highway, the car uses approximately 2 gallons per hour. Let h represent the number of hours Max has been riding on the highway. Write an expression that shows how many gallons of gasoline Max has left after driving h hours. Then find that number after Max drives 3.5 hours on the highway.
Answer:
y=18 gallons
Step-by-step explanation:
You can multiply how many hours he's been driving by the 2 gallons and subtract it from 18
Answer:
11 gallons
18=-2h h=3.5
Step-by-step explanation:
-2(3.5)=18
18-7=11, he has 11 gallons of gas left
Anne is 26 years older than her daughter, and the sum of their ages is at least 48. Which of the following inequalities, when solved, will give the possible ages of Anne's daughter?
Answer:
A = 26 + D
A + D = 48
use substitution
(26 + D) + D = 48
26 + 2D = 48
2D = 22
D = 11 years old
find the inequality that will result in the daughter being 11
Answer:
at least means greater than or equal to
Step-by-step explanation:
therefore
Question 13 options:
2x + 26 ≥ 48 ----------->this is the correct answer
x + 26 > 48
x + 26 ≥ 48
2x + 26 > 48
Sergei runs a bakery. He needs at least 175 kilograms of flour in total to complete the holiday orders he's received. He only has 34 kilograms of flour, so he needs to buy more. The flour he likes comes in bags that each contain 23 kilograms of flour. He wants to buy the smallest number of bags as possible and get the amount of flour he needs. Let F represent the number of bags of flour that Sergei buys. 1) Which inequality describes this scenario?
Answer:
Kindly check explanation
Step-by-step explanation:
Kilogram of flour needed to complete holiday order is atleast 175kg
Number of kilograms available = 34kg
Flour comes in bags each contain 23kg of flour
He wants to buy the smallest number of bags as possible and get the amount of flour he needs.
Let F = number of bags of flours Sergei needs to buy
F = (total number of kilograms needed - number of kilograms available) / kilograms per bag
F = (≥ 175 - 34) / 23
F = ≥ 141 / 23 = ≥ 6.1304347
Since flour is purchased per bag, the smallest number of bags he can possibly buy and still get the amount of flour he need = 7 bags
The price of a visit to the dentist is $ 50 $50dollar sign, 50. If the dentist fills any cavities, an additional charge of $ 100 $100dollar sign, 100 per cavity gets added to the bill. If the dentist finds n nn cavities, what will the cost of the visit be? Write your answer as an expression. $ $dollar sign .
Answer:
well you dont have a value of amounts of cavities so ill write write an equation 50+x100 where x represents amount of cavitys
A car producer stocks three types of tires: A, B, and C. Let P(A) = 0.40, P(B) = 0.15 and P(C) = 0.45. The percentage of defective tires is 2%, 1% and 5%, respectively.
Someone picks a tire off the shelf at random and it is Brand A.
If you want to know the probability that it is a defective tire (event D), which formula would you use? If you want to know the probability that it is a defective tire (event D), which formula would you use?
a) P(AD)= PD AP(A) PD)
b) P(D|A) = P(A|DP(D) P(A)
c) P(AD)= PD APD) P(A)
d) P(DA)= P(ADP(A) PD)
Answer:
The correct option is b) [tex]P(D|A)=P(A|D)P(D)[/tex].
Step-by-step explanation:
The probability of selecting the different types of tires are:
P (A) = 0.40
P (B) = 0.15
P (C) = 0.45
The defective rate for the different types of tires are:
P (D|A) = 0.02
P (D|B) = 0.01
P (D|C) = 0.05
The formula to compute the probability that the tire is defective given that it is Brand A tire as follows:
[tex]P(D|A)=P(A|D)P(D)[/tex]
If the measures of two angles add up to 180°, then the angles are supplementary.
Use the Counting Principle to find the probability. rolling a 4 on each of 4 number cubes
Answer:
Step-by-step explanation:
There is a 1/6 chance for the first cube
There is a 1/6 chance for the second cube
There is a 1/6 for the third cube
There is a 1/6 for the fourth cube.
=========
Probability for all events is (1/6)^4
P(all 1/6) = 1/1296 = 0.00077
Steve's scores on 6 of his tests were 92, 78, 86, 92, 95, and 91. If he took a seventh test and raised the mean of his scores by exactly 1 point, what was the score on the seventh test? Show your work.
Answer:
96
Step-by-step explanation:
First we need to know the mean of the Steve's scores on 6 of his tests. Given the six scores as 92, 78, 86, 92, 95, and 91.
Mean = sum of the scores/Total test taken
Mean = 92+78+86+92+95+ 91/6
Mean = 534/6
Mean = 89
If he took the seventh test and the mean score is raised by 1 them the new mean will be expressed as;
New mean = 92+78+86+92+95+ 91+x/7 = 89+1
Where x is the new score. Note that of a new score is added, the total year taken will also change to 7
To get x;
92+78+86+92+95+ 91+x/7 = 89+1
92+78+86+92+95+ 91+x/7 = 90
534+x/7 = 90
Cross multiply
533+x = 90×7
533+x = 630
x = 630-534
x = 96
Hence the score of the seventh test is 96