14. The Elizabeth Tower is 320 feet tall. At what time or times during your ride on the London Eye are you at the same height as the top of the tower? Show your work. (4 points: 2 points for finding the correct time(s), 2 points for work shown)
Answer:
Ok so on a clock there is 12 numbers where 12 is on top so at 12 am and 12 pm noon and midnight you will be at the top of the clock
Hope This Helps!!!
During the ride on the London Eye, you will be at the same height as the top of the Elizabeth Tower at approximately 21 minutes and 43.16 seconds after the start of the ride.
To determine the time(s) during the ride on the London Eye when you are at the same height as the top of the Elizabeth Tower (commonly known as Big Ben), we need to consider the height of the London Eye and its rotational motion.
Given that the Elizabeth Tower is 320 feet tall, we need to find the position of the London Eye when its height aligns with the top of the tower.
The London Eye has a height of 443 feet, and it completes one full rotation in approximately 30 minutes (or 1800 seconds). This means that it moves at a constant angular velocity of 360 degrees per 1800 seconds.
To find the time(s) when the heights align, we can set up a proportion:
(Height of the Elizabeth Tower) / (Height of the London Eye) = (Angle covered by the London Eye) / 360 degrees
Substituting the given values:
320 / 443 = (Time to align) / 1800
Simplifying the equation:
(Time to align) = (320 / 443) * 1800
Calculating the value:
(Time to align) ≈ 1303.16 seconds
Converting the time to minutes and seconds:
(Time to align) ≈ 21 minutes and 43.16 seconds
Therefore, during the ride on the London Eye, you will be at the same height as the top of the Elizabeth Tower at approximately 21 minutes and 43.16 seconds after the start of the ride.
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If the domain of a function that is translated down 3 is (0, 4), (-5, 8), (4, -2), what is the range?
A. (0, 1), (-5, 5), (4, -5)
B. (3, 4), (-2, 8), (7, -2)
C. (-3, 4), (-8, 8), (1, -2)
D. (0, 7), (-5, 11), (4, 1)
Given:
The domain of function that is translated down 3 is (0, 4), (-5, 8), (4, -2).
To find:
The range of the function.
Solution:
If a function is translated 3 units down, then
[tex](x,y)\to (x,y-3)[/tex]
Using this rule, we get
[tex](0,4)\to (0,4-3)[/tex]
[tex](0,4)\to (0,1)[/tex]
Similarly,
[tex](-5,8)\to (-5,5)[/tex]
[tex](4,-2)\to (4,-5)[/tex]
The range of the given function is (0, 1), (-5, 5), (4, -5).
Therefore, the correct option is A.
Find the missing term in the following pattern.
1984, 992, 496, blank space, 124, 62
Answer:
248
Step-by-step explanation:
common ratio for two consecutive terms is 2/1
for eg: 1984÷992 =2
992÷ 496 = 2
124÷ 62 = 2
that means 124 ×2 = 248 Answer
The following table shows the distribution of people in a tennis tournament, and one
person is to be selected at random.
Find the probability that the selected person is a female.
Express your answer as a decimal, rounded to the nearest hundredth.
Under Age 35
Male 8 Female 18
35 years and older
Male 11 Female18
Answer:
36/55
Step-by-step explanation:
Total 55 persons, total females 36.
The probability that the selected person is a female from the given table is gotten as; 0.65
What is the Probability?
From the given table we see that;
Males under 35 years = 8
Females under 35 years = 18
Males 35 years and older = 11
Females 35 years and older = 18
Thus;
Total number of people = 8 + 18 + 11 + 18
Total people = 55
Thus, probability that the selected person is a female is;
P(female) = (18 + 18)/55
P(female) = 36/55
P(female) = 0.65
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A merchant keeps marble in a cylindrical plastic container that has a diameter of 28cm and height of 35cm. A marble has a diameter of 25mm. Determine the number of marbles that can be stored in such a container if air space accounts for 20% between marbles.
Answer:
2107 marbles can be stored in the container.
Step-by-step explanation:
Since a merchant keeps marble in a cylindrical plastic container that has a diameter of 28cm and height of 35cm, and a marble has a diameter of 25mm, to determine the number of marbles that can be stored in such a container if air space accounts for 20 % between marbles, the following calculation must be performed, knowing that the volume of a cylinder is equal to height x π x radius²:
35 x 3.14 x (28/2) ² = X
109.9 x (14 x 14) = X
109.9 x 196 = X
21,540.4 = X
In turn, the volume of each 25mm diameter marble is equal to:
25mm = 2.5cm
4/3 x 3.14 x 1.25³ = X
4.18666 x 1.953125 = X
8.1770 = X
21,540.4 x 0.8 = 17,232.32
17,232.32 / 8,177 = 2,107.41
Therefore, 2107 marbles can be stored in the container.
How would I solve the question below? In what order would I solve it?
4 ⋅ 3 + 2 ⋅ 9 − 40
Step-by-step explanation:
You would multiply 4 and 3, and 2 and 9 separately, then add them, then subtract 40. Remember PEMDAS.
(4*3) + (2*9) - 40
12 + 18 - 40
-10
Hope that helps
You bought a car that was $25500 and the value depreciates by 4.5% each year.
How much will the car be worth after 5 years?
How much after 8 years?
Answer:
(a) 20256.15625
(b) 17642.78546
Step-by-step explanation:
(a) There's a formula for this problem y = A(d)^t where, A is the initial value you are given, d is the growth or decay rate and t is the time period. So, in this case, as the car cost is decreasing it is a decay problem and we can write the formula as such; y = A(1-R)^t
So, in 5 years the car will be worth, 25500(1-4.5%)^5 or 20256.15625 dollars
(b) And after 8 years the car will be worth 25500(1-4.5%)^8 or 17642.78546 dollars.
Solve the equation for x.
2/3x-1/9x+5=20
Answer:
x = 27
Step-by-step explanation:
I'm assuming the equation looks like this:
[tex]\frac{2}{3}x-\frac{1}{9}x+5=20[/tex]
Here's how to solve for x:
[tex]\frac{2}{3}x-\frac{1}{9}x+5=20[/tex]
(subtract 5 from both sides)
[tex]\frac{2}{3}x-\frac{1}{9}x=15[/tex]
(Find the GCF of 3 and 9, which is 3. Multiply 2/3 by 3/3. You get 6/9)
[tex]\frac{6}{9}x-\frac{1}{9}x=15[/tex]
(add like terms)
[tex]\frac{5}{9}x=15[/tex]
(multiply 9/5 to both sides, which is the same as dividing both sides by 5/9)
x = 27
Hope it helps (●'◡'●)
A boy had 3 apples and lost one, how many does he have now
Step-by-step explanation:
i would love to say 2 but the word had shows that he does not have 3 apples anymore so the answer is either
0 or -1
The number of apples left after taking the 1 apple from 3 apples by a person is 2 apples.
What is subtraction?Subtraction stands for the resultant number, which exists acquired by taking the difference of a number from another number.
Let a number be subtracted from the number b. Then the consequent number after subtracting b from a will be,
d = b - a
Here, (a, b) exists the real numbers.
It exists given that there exist 3 apples. 1 apple stand was taken. Let's assume after taking the 3 apples, that there exist x apples remaining.
As there exist a total of 3 apples and 1 apple stand taken, then to estimate the number of apples left, we must subtract 1 apple from 3 apples.
Therefore, the total apples left exist,
x = 3 - 1
x = 2
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The running trail in the local park is 2.826 miles long. If the park board were planning to extend the trail by 1.46 miles, what would the new length of the running trail be?
Answer:
4.286
Step-by-step explanation:
you really need help with this ? you cannot just use your calculator ? that would have been faster than putting that question in here ...
remember, similar to the number positions in front of the decimal point, it is equally important to add the same positions after the decimal point.
we have 10th, 100th, 1000th, 10000th, 100000th, ... no end possible.
so we have
2.826 miles
and need to add 1.46 miles
2.826
1.46
----------
4.286
and the line of thinking goes from right to left
nothing plus 6 is 6
6 plus 2 is 8
4 plus 8 is 12, so we write 2 and carry over the 1
1 plus 2 plus 1 carry over is 4
if it helps, you can always add zeroes at the end of any digits after the decimal point, as you can also add zeroes in front to the digits before the decimal point to make both numbers have the same length and their decimal points are perfectly aligned.
our addition could have also looked like
2.826
1.460
with the same result
overall, if this is truly helping you, an example of using both leading and tailing zeroes could be
4278.9472081
0021.6380000
---------------------
4300.5852081
Thirty-six percent of customers who purchased products from an e-commerce site had orders exceeding 110. If 17% of customers have orders exceeding 110 and also pay with the e-commerce site's sponsored credit card, determine the probability that a customer whose order exceeds 110 will pay with the sponsored credit card.
Answer:
The right solution is "0.5".
Step-by-step explanation:
According to the question,
P(pay with the sponsored credit card | order exceeds $110)
= [tex]\frac{P(Pay \ with \ the \ sponsored \ credit\ card\ and\ order\ exceeds\ 110)}{P(order \ exceeds \ 110)}[/tex]
= [tex]\frac{P(A \ and \ B)}{P(A)}[/tex]
By putting the values, we get
= [tex]\frac{0.17}{0.34}[/tex]
= [tex]0.5[/tex]
Thus, the above is the right solution.
3. The size of a red blood cell is 0.000007 m and the size of a plant
cell is 0.0000127 m. Compare these two.
Given:
Size of a red blood cell = 0.000007 m
Size of a plant cell = 0.0000127 m
To find:
The comparison of these two values.
Solution:
We have,
Size of a red blood cell = 0.000007 m
Size of a plant cell = 0.0000127 m
Clearly, [tex]0.0000127>0.000007[/tex]. Now, the difference between these two values is:
[tex]0.0000127-0.000007=0.0000057[/tex]
Therefore, the size of a plant cell is 0.0000057 m more than the size of a red blood cell.
Assume that the breaking system of a train consists of two components connected in series with both of them following Weibull distributions. For the first component the shape parameter is 2.1 and the characteristic life is 100,000 breaking events. For the second component the shape parameter is 1.8 and characteristic life of 80,000. Find the reliability of the system after 2,000 breaking events:
Answer:
0.9984
Step-by-step explanation:
we have shape parameter for the first component as 2.1
characteristics life = 100000
for this component
we have
exp(-2000/100000)².¹
= e^-0.0002705
= 0.9997
for the second component
shape parameter = 1.8
characteristic life = 80000
= exp(-2000/80000)¹.⁸
= e^-0.001307
= 0.9987
the reliability oif the system after 2000 events
= 0.9987 * 0.9997
= 0.9984
A textile fiber manufacturer is investigating a new drapery yarn, which the company claims has a mean thread elongation of 12 kilograms with a standard deviation of 0.5 kilograms. The company wishes to test the hypothesis H0: µ = 12 against H1: µ < 12 using a random sample of n = 4 specimens. Calculate the P-value if the observed statistic is Xbar (average) = 11.25. Suppose that the distribution of the sample mean is approximately normal.
Answer:
The p-value of the test is 0.0013.
Step-by-step explanation:
The test statistic is:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.
12 is tested at the null hypothesis:
This means that [tex]\mu = 12[/tex]
Standard deviation of 0.5 kilograms.
This means that [tex]\sigma = 0.5[/tex]
Sample of n = 4 specimens. Observed statistic is Xbar (average) = 11.25.
This means that [tex]n = 4, X = 11.25[/tex]
Value of the test statistic:
[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \frac{11.25 - 12}{\frac{0.5}{\sqrt{4}}}[/tex]
[tex]z = -3[/tex]
P-value:
Probability of finding a sample mean belo 11.25, which is the p-value of z = -3.
Looking at the z-table, z = -3 has a p-value of 0.0013, thus the this is the p-value of the test.
Sally bought five books.Their mean price was 3.25. The total cost for four books was 11.75.what was the cost of the fifth book
Answer:
$4.50
Step-by-step explanation:
Use the mean formula: mean = sum of elements / number of elements
Let x represent the cost of the fifth book, and solve for x:
mean = sum of elements / number of elements
3.25 = (11.75 + x) / 5
16.25 = 11.75 + x
4.5 = x
So, the cost of the fifth book was $4.50
Suppose that the probability distribution for birth weights is normal with a mean of 120 ounces and a standard deviation of 20 ounces. The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is [ Select ] 68%. The probability that a randomly selected infant has a birth weight between 110 and 130 is [ Select ] 68%.
Answer:
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is 68%.
The probability that a randomly selected infant has a birth weight between 110 and 130 is 38%.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of 120 ounces and a standard deviation of 20 ounces.
This means that [tex]\mu = 120, \sigma = 20[/tex]
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is
p-value of Z when X = 140 subtracted by the p-value of Z when X = 100.
X = 140
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{140 - 120}{20}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.84
X = 100
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{100 - 120}{20}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a p-value of 0.16
0.84 - 0.16 = 0.68
The probability that a randomly selected infant has a birth weight between 100 ounces and 140 ounces is 68%.
The probability that a randomly selected infant has a birth weight between 110 and 130
This is the p-value of Z when X = 130 subtracted by the p-value of Z when X = 110.
X = 130
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{130 - 120}{20}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a p-value of 0.69
X = 110
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{110 - 120}{20}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a p-value of 0.31
0.69 - 0.31 = 0.38 = 38%.
The probability that a randomly selected infant has a birth weight between 110 and 130 is 38%.
Can the range of a function be written like this {6,7,8,10} instead of like this [tex]6\leq x\leq 10[/tex]?
Answer:
No unless x is being used to define only elements of an integer set.
Step-by-step explanation:
No, not in general unless x is defined as a integer or a subset of the integers like the naturals, whole numbers....
Usually 6<=x<=10 means all real numbers between 6 and 10, inclusive. This means example that 6.6 or 2pi are in this set with infinitely other numbers that I can't name.
{6,7,8,9,10} just means the set containing the numbers 6,7,8,9,10 and that's only those 5 numbers.
What is the smallest 6-digit palindrome (a number that reads the same forward and
backward) divisible by 99?
Answer:
108801
Step-by-step explanation:
Well, you should first add 99 to 99999 which is 10098. And since it's not a palindrome you need to keep adding 99 to the sum until you reach one.
----------------------------------
This is with a calculator
Btw, I used calculator soup.com for it.
100089, 100188, 100287, 100386, 100485, 100584, 100683, 100782, 100881, 100980, 101079, 101178, 101277, 101376, 101475, 101574, 101673, 101772, 101871, 101970, 102069, 102168, 102267, 102366, 102465, 102564, 102663, 102762, 102861, 102960, 103059, 103158, 103257, 103356, 103455, 103554, 103653, 103752, 103851, 103950, 104049, 104148, 104247, 104346, 104445, 104544, 104643, 104742, 104841, 104940, 105039, 105138, 105237, 105336, 105435, 105534, 105633, 105732, 105831, 105930, 106029, 106128, 106227, 106326, 106425, 106524, 106623, 106722, 106821, 106920, 107019, 107118, 107217, 107316, 107415, 107514, 107613, 107712, 107811, 107910, 108009, 108108, 108207, 108306, 108405, 108504, 108603, 108702, 108801, 108900, 108999, 109098, 109197, 109296, 109395, 109494, 109593, 109692, 109791, 109890
HELP AGAIN
235 ≤-8(1+5x)+3
i need the steps as well
Answer:
x ≤ -6
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightEquality Properties
Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of EqualityStep-by-step explanation:
Step 1: Define
Identify
235 ≤ -8(1 + 5x) + 3
Step 2: Solve for x
[Subtraction Property of Equality] Subtract 3 on both sides: 232 ≤ -8(1 + 5x)[Division Property of Equality] Divide -8 on both sides: -29 ≥ 1 + 5x[Subtraction Property of Equality] Subtract 1 on both sides: -30 ≥ 5x[Division Property of Equality] Divide 5 on both sides: -6 ≥ xRewrite: x ≤ -6Step-by-step explanation:
To solve for x, make sure you move everything else to the other side of the ≤ sign.
So,
[tex]235\leq -8(1+5x)+3\\232\leq -8-40x\\240\leq -40x\\-6\geq x[/tex]
* Remember that the sign changes anytime you divide by a negative number!
So your answer is:
[tex]x\leq -6[/tex], x is less than or equal to -6.
3. Tell whether each statement is true or false Explain how you know a) LCM (7, 18) - LCM (14.18) b) LCM (5,8) - LCM (10,8) c) The GCF of any two prime numbers is 1 and the number itself.
Step-by-step explanation:
ok for a. the both are 126
and for b. the both are 30
for c. i believe its true
Solve (x - 5)2 = 3.
Answer:
x = 5±√3
Step-by-step explanation:
Equation: (x-5)² = 3
Step 1: Take the square root of both side of the equation
√(x-5)² = ±√3
x-5 = ±√3
Step 2: add 5 to both side of the equation
x-5+5 = 5±√3
x = 5±√3
Hence, from the options above, the right answer is
B. x = 5±√3
PLEASE ANSWER ILL MARK !!
Step-by-step explanation:
a) Use sine law:
[tex]\dfrac{g}{\sin 60} = \dfrac{17\:m}{\sin 49}[/tex]
Solving for g,
[tex]g = \left(\dfrac{\sin 60}{\sin 49}\right)(17\:m)=19.5\:m[/tex]
b) Use the cosine law here:
[tex]q^2 = (11\:\text{cm})^2 + (16\:\text{cm})^2 \\ - 2(11\:\text{cm})(16\:\text{cm})\cos 29[/tex]
Solving for q,
[tex]q = 8.3\:\text{cm}[/tex]
Uma pizzaria oferece em seu cardápio 12 sabores de pizza. Se um cliente pretende pedir 3 pizzas, então o número de maneiras que ele pode realizar esse pedido é;
•364
•220
•440
•1320
Answer:
Step-by-step explanation:
Partindo do pressuposto de que você pode ter coberturas duplas e triplas do mesmo item, o cálculo é relativamente simples. Para calcular as combinações possíveis; deve-se multiplicar as coberturas disponíveis pelo número total de coberturas permitidas. Este cálculo é semelhante a como olhamos para diferentes sistemas de contagem de base. Normalmente contamos com decimais (base 10), portanto, o número de combinações, se usar 3 dígitos, seria calculado por 10 x 10 x 10.
10x10 = 100
100x10 = 1000 combinações (0 a 999)
Sua pergunta sobre coberturas de pizza é a mesma, mas assumindo um sistema de numeração de base 12, então 12x12x12 ou 12³
Portanto, 1.728 combinações incluindo 0 (sem coberturas?) E também incluindo 12 ocasiões em que todas as 3 coberturas seriam iguais. Se esses cenários de pessoas forem restritos de modo que você só possa ter coberturas duplas máximas, etc., então essas combinações devem ser removidas (subtraídas do total de combinações permitidas).
Espero ter ajudado você a entender os princípios, então você deve ser capaz de trabalhar a partir disso, de muitas outras soluções semelhantes
In a survey, 250 adults and children were asked whether they know how to
swim. The survey data are shown in the relative frequency table.
Can swim
Cannot swim
Total
0.34
Adults
Children
0.06
0.48
0.12
Total
What percentage of the people surveyed can swim?
O A. 18%
B. 82%
C. 48%
D. 34%
Answer:
B - 82%
Step-by-step explanation:
.34+.48
The percentage of people who can swim is 82%.
Option B is the correct answer.
What is a percentage?
The percentage means the required value out of 100.
It is calculated by dividing the required value by the total value and multiplying it by 100.
The percentage change is also calculated using the same method.
In percentage change, we find the difference between the values given.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
We have,
The relative frequency table shows the proportion of people in each group who can and cannot swim.
To find the percentage of people who can swim, we need to add up the proportion of adults who can swim (0.34) and the proportion of children who can swim (0.48).
Percentage of people who can swim
= (0.34 + 0.48) x 100%
= 82%
Therefore,
The percentage of people who can swim is 82%.
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I’m stuck please help .
Answer:
me too
Step-by-step explanation:
me too
Solve algebraically.
6(t-2) + 15t < 5(5 + 3t)
With work shown please!!
Step-by-step explanation:
6t-12+15t | 25+15t
21t-12 | 25+15t
21t-12 < 25+15t
hence proved..
Answer:
21t - 12 < 25 + 15t
Step-by-step explanation:
6( t - 2 ) + 15t < 5 ( 5 + 3t )
Distribute .6t - 12 + 15t < 25 + 15t
Combine like terms.21t - 12 < 25 + 15t.
Hence , Proved.
al of
10. A square field has four sprinklers that spray
in the areas represented by the circles below. If
the shaded portion represents area that is not
reached by the sprinklers, find the total area that
is not reached by the sprinklers.
Using the areas of the sqaure and of the circle, it is found that the total area that is not reached by the sprinklers is of 343.36 ft².
What is the area of a square?The area of a square of side length l is given by:
A = l²
In this problem, we have that l = 40 ft, hence:
A = (40 ft)² = 1600 ft².
What is the area of a circle?The area of a circle of radius r is given by:
[tex]A = \pi r^2[/tex]
In this problem, we have four circles of radius r = 10 ft, hence it's combined area, in square feet, is given by:
[tex]A_c = 4\pi (10)^2 = 400\pi = 1256.64 \text{ft}^2[/tex]
The area not reached by the sprinklers is the subtraction of the area of the square by the area of the circle, hence:
1600 - 1256.64 = 343.36 ft².
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What are the solutions of the quadratic equation 49x2 = 9?
A. x = 1/9 and x = -1/9
B. x = 3/7 and x = -3/7
C. x = 3/4 and x = -3/4
D. x = 9/49 and x = -9/49
Brainliest if you explain how. got stumped on this one
Answer:
B
Step-by-step explanation:
49x^2=9
solve for x
x^2= 9/49
x=± [tex]\sqrt{9/49\\}[/tex]
which is x = ±3/7 (B)
Answer: b x=1/9 and x=-1/9
Step-by-step explanation:
Find dy/dx of the function y = √x sec*-1 (√x)
Hi there!
[tex]\large\boxed{\frac{dy}{dx} = \frac{1}{2\sqrt{x}}sec^{-1}(\sqrt{x}) + \frac{1}{2|\sqrt{x}|\sqrt{{x} - 1}}}[/tex]
[tex]y = \sqrt{x} * sec^{-1}(-\sqrt{x}})[/tex]
Use the chain rule and multiplication rules to solve:
g(x) * f(x) = f'(x)g(x) + g'(x)f(x)
g(f(x)) = g'(f(x)) * 'f(x))
Thus:
f(x) = √x
g(x) = sec⁻¹ (√x)
[tex]\frac{dy}{dx} = \frac{1}{2\sqrt{x}}sec^{-1}(\sqrt{x}) + \sqrt{x} * \frac{1}{\sqrt{x}\sqrt{\sqrt{x}^{2} - 1}} * \frac{1}{2\sqrt{x}}[/tex]
Simplify:
[tex]\frac{dy}{dx} = \frac{1}{2\sqrt{x}}sec^{-1}(\sqrt{x}) + \sqrt{x} * \frac{1}{2|x|\sqrt{{x} - 1}}[/tex]
[tex]\frac{dy}{dx} = \frac{1}{2\sqrt{x}}sec^{-1}(\sqrt{x}) + \frac{1}{2|\sqrt{x}|\sqrt{{x} - 1}}[/tex]
Answer:
[tex]\displaystyle y' = \frac{arcsec(\sqrt{x})}{2\sqrt{x}} + \frac{1}{2|\sqrt{x}|\sqrt{x - 1}}[/tex]
General Formulas and Concepts:
Algebra I
Exponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]Exponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Derivatives
Derivative Notation
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Arctrig Derivative: [tex]\displaystyle \frac{d}{dx}[arcsec(u)] = \frac{u'}{|u|\sqrt{u^2 - 1}}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle y = \sqrt{x}sec^{-1}(\sqrt{x})[/tex]
Step 2: Differentiate
Rewrite: [tex]\displaystyle y = \sqrt{x}arcsec(\sqrt{x})[/tex]Product Rule: [tex]\displaystyle y' = \frac{d}{dx}[\sqrt{x}]arcsec(\sqrt{x}) + \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})][/tex]Chain Rule: [tex]\displaystyle y' = \frac{d}{dx}[\sqrt{x}]arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{d}{dx}[\sqrt{x}] \bigg][/tex]Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle y' = \frac{d}{dx}[x^\bigg{\frac{1}{2}}]arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{d}{dx}[x^\bigg{\frac{1}{2}}] \bigg][/tex]Basic Power Rule: [tex]\displaystyle y' = \frac{1}{2}x^\bigg{\frac{1}{2} - 1}arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{1}{2}x^\bigg{\frac{1}{2} - 1} \bigg][/tex]Simplify: [tex]\displaystyle y' = \frac{1}{2}x^\bigg{\frac{-1}{2}}arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{1}{2}x^\bigg{\frac{-1}{2}} \bigg][/tex]Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle y' = \frac{1}{2x^\bigg{\frac{1}{2}}}arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{1}{2x^\bigg{\frac{1}{2}}} \bigg][/tex]Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle y' = \frac{1}{2\sqrt{x}}arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{d}{dx}[arcsec(\sqrt{x})] \cdot \frac{1}{2\sqrt{x}} \bigg][/tex]Arctrig Derivative: [tex]\displaystyle y' = \frac{1}{2\sqrt{x}}arcsec(\sqrt{x}) + \bigg[ \sqrt{x}\frac{1}{|\sqrt{x}|\sqrt{(\sqrt{x})^2 - 1}} \cdot \frac{1}{2\sqrt{x}} \bigg][/tex]Simplify: [tex]\displaystyle y' = \frac{arcsec(\sqrt{x})}{2\sqrt{x}} + \frac{1}{2|\sqrt{x}|\sqrt{x - 1}}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
What is the correct line graph for y=3x+5?
Answer:
The equation y=−3x+5 is in slope intercept form, and represents a straight line in which -3 is the slope, and 5 is the y -intercept.
