The probability for success is not the same for each trial, then the option D cannot be modelled using a geometric distribution.
What is probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
One of the assumptions to consider before the use of a geometric distribution can be valid is that, the probability of success must be the same for each outcome.
In the option D, selection is done without replacement, meaning that, the number total possible outcome will decrease as we make each selection and will also depend on how the colours are being chosen.
Hence, the probability of success will differ for each outcome, thus the option D is correct.
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Identify the range of y = x² - 2x - 3 if the domain is all real numbers.
The range of this quadratic function y = x² - 2x - 3 is all real numbers greater than 2.
What is the domain and range of a function?Suppose we have an ordered pair (x, y) then the domain of the function is the set of values of x and the range is the set of values of y for which x is defined.
We know a quadratic function y = ax² + bx + c is a graph of a parabola that opens upwards as the coefficient of x² is positive.
Given, y = x² - 2x - 3.
y = x² - 3x + x - 3.
y = x(x - 3) + 1(x - 3).
y = (x - 3)(x + 1).
Now the x-intercepts are at 3 and -1 so the vertex is located at (3 - 1)/2 = 2.
So the range of this function is all real numbers greater than 2 as the parabola opens upwards.
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8y-2x=-40 put the following equation in slope intercept form simplyifing all fractions
Answer:
[tex]y = \frac{1}{4} x\\[/tex] [tex]- 5[/tex]
Step-by-step explanation:
Take a look at the attachment...
Hope this helps! :)
C and D are sets of real numbers defined as follows. Write using interval notation.
The given sets written in interval notation are
C = (4, ∞), and = (-∞, 9]
Writing sets in interval notationFrom the question, we are to write the given sets using the interval notation
The given sets are
C = {v | v > 4}
D = {v | v ≤ 9}
For set C,
C = {v | v > 4}
The elements of the sets are 5, 6, 7, 8, 9, ...
This can be written in interval notation as (4, ∞)
For set D,
D = {v | v ≤ 9}
The elements of the sets are 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, ...
This can be written in interval notation as (-∞, 9]
Hence, the interval notation form of the sets are
C = (4, ∞)
and
D = (-∞, 9]
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Find the equation of the tangent line to the graph of the function f(x)=(x^2+6)(x−4) at the point (1,−21) .
im stumped and this is due soon, please help
The equation of the tangent line of the function is y = x + 20
How to find the equation of the tangent linef(x)=(x^2+6)(x−4)
expanding the function gives
f(x)=x^3 - 4x^2 + 6x - 24
At a specific point the slope, m is the derivative
finding the derivative of f(x)
f'(x) = 3x^2 - 8x + 6
The derivative at the point (1, -21) is found by substituting the point to the equation
f'(1) = 3 * 1 - 8 * 1 + 6 = 1
the slope, m = 1
equation of the line of slope = 1 passing through point (1, -21) is calculated using the point slope formula which is
(y - y') = m(x - x')
(y - -21) = 1(x - 1)
y + 21 = x - 1
y = x + 20
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The following diagram describes the volume of blue and red paint in a mixture.What is the ratio of blue paint to total paint in the mixture? Choose 1 answer:
Answer:
C
Step-by-step explanation:
blue volume = 10
red volume = 6
total = 10 6 = 16
ratio of
blue : total
= 10 : 16 ( divide both parts by 2 )
= 5 : 8
Select ALL expressions that are close to a quotient of 1.
(A) 1900 / 21
(B) 210,000 / 190,000
(C) 19 / 210
(D) 2.1 / 1.9
(E) 190 / 210
(F) 190 / 21
Answer:
Step-by-step explanation:
C E F or just E and F
Vrite an equation in slope-intercept form of the line that passes through (1, −9) and (-3, -9).
Answer:
y = -9
Step-by-step explanation:
The slope-intercept form is y = mx + b
From two points we can easily determine the slope, m = (y2-y1)/(x2-x1)
You would put in the values into the equation.
m = (-9-(-9))/(-3-1) = 0/-4 = 0
Using the slope-intercept form, y = 0(x) + b ⇒ y = b
But we don't know what the y-intercept (b) is yet. Using the point-slope form we can determine b.
Point-slope: (y - y1) = m(x - x1)
(y - (-9)) = 0(x - (-3))
y + 9 = 0
y = -9
So the equation is y = -9.
Triangle CDE is similar to triangle FGH. Find the measure of side HF. Round your
answer to the nearest tenth if necessary.
E
23
D
19
H
G
59
Answer:
HF ≈ 71.4
Step-by-step explanation:
since the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{HF}{EC}[/tex] = [tex]\frac{FG}{CD}[/tex] ( substitute values )
[tex]\frac{HF}{23}[/tex] = [tex]\frac{59}{19}[/tex] ( cross- multiply )
19 HF = 1357 ( divide both sides by 19 )
HF ≈ 71.4 ( to the nearest tenth )
The cost C (in dollars) for a company to produce and sell x thousand gadgets is given by C = 1 /30 x^2 − 2x + 2530
(a) What is the company's start-up cost? _____________ $
(b) What is the minimum cost? ______________$
(c) How many gadgets must the company produce and sell in order to incur the least cost? (Be careful with your units!) ______________ gadgets
The company's start-up cost is equal to $1/2530.
The company's minimum cost is equal to $15/37,949.
The number of gadgets that this company must produce and sell in order to incur the least cost is equal to 1/30 gadgets.
How to determine the company's start-up cost?From the information provided, the cost C (in dollars) for a company to produce and sell x thousand gadgets is modeled by the following function:
Function, C = 1/(30x² − 2x + 2530)
Mathematically, the start-up cost for this company is equal to the total cost when no sales of its gadget has been made i.e the level of activity is at zero (x = 0). Therefore, we would substitute the value of x in the function with zero (0):
Start-up cost = 1/(30(0)² − 2(0) + 2530)
Start-up cost = $1/2530.
What is the minimum cost?First of all, we would determine the number of gadgets that this company must produce and sell in order to incur the least cost by differentiating the function for the total cost using the quotient rule:
dC/dx = [(0(30x² − 2x + 2530) - 1(60x - 2))1/(30x² − 2x + 2530)²]
dC/dx = [(0 - 60x + 2)1/(30x² − 2x + 2530)²]
Equating the marginal cost function to zero, we have:
-60x + 2/(30x² − 2x + 2530)² = 0
-60x + 2 = 0
60x = 2
x = 2/60
x = 1/30 gadgets.
Now, we can determine the minimum cost:
Function, C = 1/(30x² − 2x + 2530)
Minimum cost = 1/(30(1/30)² − 2(1/30) + 2530)
Minimum cost = 1/(1/30 - 1/15 + 2530)
Minimum cost = 1/(-1/15 + 2530)
Minimum cost = 1/(37,949/15)
Minimum cost = $15/37,949
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Jack had received a collection of 30 baseball cards and decided he was going to start collecting. If Jack adds 2 stamps every month, how long will it take to have 50 stamps in his collection?
2. Bella opens a savings account with $100 and the bank pays 1% monthly interest rate. How much money will Bella have after 12 months, if she does not deposit or draw an amount?
If someone answers this question they have to determine which equation is linear and which is nonlinear. Then, write a function that models the linear situation and then write 1 ordered pair that is the solution to the situation.
1) The amount of time that it will take to have 50 stamps is; 10 months
2) The amount of money that bella will have after 12 months is; $112
How to solve linear equation word problems?1) We are told that Jack received a collection of 30 baseball cards. Now, he decided to start collecting after that and he adds 2 stamps every month. Thus, let m represent the number of months he collects and so we have the equation as;
2m + 30
Now, we want to find out how long it will take to get 50 stamps. Thus;
2m + 30 = 50
2m = 20
m = 10 months
2) Principal = $100
Interest rate = 1% monthly
Thus;
Interest after 12 months = 100(0.01 * 12) = $12
Thus, amount she now has after 12 months = $100 + $12 = $112
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At a food festival, 38 of the dishes were from China. Another 12.5% of the dishes were from Japan. What percent of the dishes were from other countries?
If 3/8 fraction of the dishes were from China and another 12.5 percent from Japan, it implies that 50 percent were from other countries.
How is the percentage determined?Firstly, we convert the proper fraction to a percentage by expressing it over 100.
Secondly, the percentages of China and Japanese dishes are added before being subtracted from 100%.
The resulting difference is the percentage of dishes from other countries.
In summary, unit conversion, expressing the fraction as a percentage, and mathematical operations of addition and subtraction are involved.
Dishes from China = 3/8 = 0.375 or 37.5% (3/8 x 100)
Dishes from Japan = 12.5%
Dishes from other countries = 50% (1 - 37.5% - 12.5%)
Thus, the percentage of dishes from other countries is 50%.
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Question Completion:At a food festival, 3/8 of the dishes were from China. Another 12.5% of the dishes were from Japan.
How many minutes will it take to burn 110 calories
Consider the following scenario: P(E) = 0.4, P(F) = 0.5
a. Find P(E or F) if E and F are disjoint
b. Find P(E or F) if E and F are independent
The probabilities are given as follows:
a. P(E or F) if E and F are disjoint: 0.9 = 90%.
b. P(E or F) if E and F are independent: 0.7 = 70%.
What is a Venn probability?In a Venn probability, two events are related with each other, as are their probabilities.
The "or probability" is given by:
[tex]P(A \cup B) = P(A) + P(B) - P(A \cap B)[/tex]
Disjoint events are events that cannot happen at the same time, hence P(E and F) = 0, and then the probability is:
P(E or F) = P(E) + P(F) = 0.4 + 0.5 = 0.9 = 90%.
When E and F are independent, we have that:
P(E and F) = P(E)P(F) = 0.4 x 0.5 = 0.2.
Hence the probability in item b is given by:
P(E or F) = P(E) + P(F) - P(E and F) = 0.4 + 0.5 - 0.2 = 0.7 = 70%.
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North Dakota has 465 campgrounds over 12 counties while West Virginia has 711 campgrounds. If North Dakota was proportional to West Virginia in the number of campgrounds to counties, how many counties would West Virginia be expected to have? Round to the nearest whole number.
Number of countries would west Virginia be expected to have is 18 countries
The number of campgrounds that North Dakota has = 465 campgrounds
The number of countries = 12 countries
The number of campgrounds that West Virginia has = 711 campgrounds
Given that the North Dakota was proportional to West Virginia in the number of campgrounds to counties.
The proportion will be
465 / 12 = 711 / x
Divide the terms
38.75 = 711 / x
x = 711 / 38.75
Divide the terms
x = 18.4
x ≈ 18 countries
Hence, number of countries would west Virginia be expected to have is 18 countries
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A toy rocket is shot vertically into the air from a launching pad 7 feet above the ground with an initial velocity of 72 feet per second. The height h, in feet, of the rocket above the ground at t seconds after launch is given by the function h(t)=-16 t²+72 t+7. How long will it take the rocket to reach its maximum height? What is the maximum height?
The rocket reaches its maximum height at ? second(s) after launch.
(Simplify your answer.)
The maximum height reached by the object is ? feet.
(Simplify your answer.)
Answer:
The rocket reaches its maximum height at 2.25 second(s) after launch.
The maximum height reached by the object is 88 feet.
Step-by-step explanation:
How long will it take the rocket to reach its maximum height?
h(t)=-16t² + 72t +7 is a quadratic equation in the form at²+bt+c, where a=-16 and b=72 and c=7.
The maximum height would be at the vertex / peak of this equation, and we can find the x-coordinate (aka the value of 't') at this point using the formula [tex]\frac{-b}{2a}[/tex]:
[tex]t=\frac{-b}{2a} =\frac{-72}{2(-16)} =\frac{72}{32} =2.25seconds[/tex]
What is the maximum height?
Since we know how long it takes to reach maximum height, we can sub this value into the equation to find our answer:
h(t)=-16t² + 72t +7
h(2.25)=-16(2.25)² + 72(2.25) +7
=-81+162+7=88feet
There are two number cubes (dice) with faces labeled 1, 2, 3, 4, 5 and 6 equally likely. Roll two dice (one at a time) and calculate the sum of the two faces.
a. Display the sample space
b. P(sum of 8) =
c. P(odd sum or sum greater than 9)
d. P(odd sum and sum greater than 9)
Let A = rolling a 3 or a 4 first
Let B = sum of at most 7
Are A and B mutually exclusive? Why or why not?
Are A and B independent? Why or why not?
a. Please find attached the sample space
b. P(sum of 8) = 5/36
c. P(odd sum or sum greater than 9) = 2/3
d. P(odd sum and sum greater than 9) = 1/12
e. P(A) = 1/3
f. P(B) = 1/6
g. P(A and B) = 1/18
h. A and B are not mutually exclusive because both events can occur at the same time.
i. A and B are not independent because the occurrence of A and lead to the occurrence of B.
What are the probabilities?P(sum of 8) = number of sample spaces that sum up to 8 / total number of sample spaces
P(sum of 8) = 5/36
P(odd sum or sum greater than 9) = (sample spaces that have an odd sum / total number of sample spaces) + (sample spaces that have a sum greater than 9 / total number of sample spaces)
P(odd sum or sum greater than 9) =(18 / 36) + (6/36) = 24/36 = 2/3
P(odd sum and sum greater than 9) = (sample spaces that have an odd sum / total number of sample spaces) x (sample spaces that have a sum greater than 9 / total number of sample spaces)
P(odd sum and sum greater than 9) = 18/36 x 6/36 = 1/12
P(A) = (sample spaces that have a 3 first / total number of sample spaces) + (sample spaces that have a 3 first / total number of sample spaces)
P(A) = 6/36 + 6/36 = 1/3
P(B) = (sample spaces that adds up to 7 / total number of sample spaces)
= 6/36 = 1/6
P(A and B) = 1/3 x 1/6 = 1/18
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An open box is made from a 40 -cm by 50-cm piece of tin by cutting a square from each corner and folding the edges. The area of the resulting base is 1496cm squared. What is the length of the sides of the squares?
The length of the sides of the squares is 3 cm.
What is a rectangle?A rectangle is a quadrilateral with all four interior angles 90°.
Given that, the dimension of the rectangular tin piece is 40 cm by 50 cm.
Let the length of the sides of the square is x.
Then, the dimensions of the resulting rectangular base are (40 - 2x) by
(50 -2x).
Given that, the area of the resulting base is 1496 square cm.
Therefore, it follows:
(40 - 2x)(50 -2x) = 1496
2000 - 80x -100x + 4x² = 1496
4x² - 180x + 2000 -1496 = 0
4x² - 180x + 504 = 0
x² - 45x + 126 = 0
x² - 42x - 3x + 126 = 0
x(x- 42) - 3(x - 42) = 0
(x - 3)(x - 42) = 0
x-3 =0 or x - 42 = 0
x = 3 or x = 42
Note that the dimension of the rectangular tin piece is 40 cm by 50 cm, therefore, x = 42 cm cannot be the length of the square.
Hence, the length of the sides of the squares is 3 cm.
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There are 30 apples in a box.
The mean weight of these 30
apples is 99 grams.
There are 25 apples in a bag.
The mean weight of the
apples in the box and in the
bag combined is 94 grams.
Work out the mean weight
of the 25 apples in the bag.
Answer: 88 grams
Step-by-step explanation:
The total weight of the 30 apples is [tex](30)(99)=2970[/tex] grams.
The total weight of all the apples in the box and bag is [tex]55(94)=5170[/tex] grams.
So, the weight of the apples in the bag is [tex]5170-2970=2200[/tex] grams.
Thus, the mean weight is [tex]\frac{2200}{25}=88[/tex] grams.
Find a polynomial f(x) of degree 4 that has the following zeros.
-3, 4, 0, 8
Leave your answer in factored form.
f(x) = 0
Polynomial of degree 4 in factored form is x(x+3)(x-4)(x-8) = 0
The Standard form of polynomial with degree 4 is ax⁴ + bx³ + cx² + dx + e
Zeros of the polynomial is the points where the polynomial becomes zero.
According to the question,
The zeros of the polynomial are -3 , 4 , 0 , 8
Let equation of polynomial f(x) be
f(x) = ax⁴ + bx³ + cx² + dx + e ------------------(1)
Sum of zeros = -b/a = -3 + 4 + 0 + 8
= 9
Sum of product of two zeros = c/a = (-3× 4 ) +(0 × 4) + (0 × 8) + (8× -3)
= -12 + 0 + 0 -24
= -12 - 24
= -36
Sum of product of sum of three zeros = -d/a = ( -3×4×0) + ( 4×0 ×8 ) + (0×8×-3) + (-3×4×8)
= 0 - 96
= -96
Product of all four zeros = e/a = (-3×4×0×8)
= 0
So, b = -9 , c =-36 , d = 96 and e = 0
Substituting value in equation (1),
Hence , the required polynomial = x⁴ -9x³ -36x²+96x
Factored form ,
x⁴ -9x³ -36x² +96x = 0
=> x(x³ - 9x² - 36x + 96) = 0
=> x(x+3)(x-4)(x-8) = 0
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Solve the equation √3 Cosecx-2=0
Answer:
60
Step-by-step explanation:
[tex] \sqrt{3} cosecx - 2 = 0[/tex]
[tex] \sqrt{3} cosecx = 2[/tex]
[tex]cosec = \frac{2}{ \sqrt{3} } [/tex]
[tex]sinx = \frac{ \sqrt{3} }{2} [/tex]
[tex]x = sin {?}^{ - 1} ( \frac{ \sqrt{3} }{2} )[/tex]
[tex]x = 60[/tex]
Mrs. Smit has a yearly budget of 2.300 for the math club. she spends 36% of the budget for transportation. snacks . and rewards. 70% of the 36% was used for transportation. more money was spent on snacks iban rewards
complete the table so show possible amounts of money. in dollars Mrs . Smith could have spent this year in each category
1,858.40
2300-36%=1472
1472-70%=441.60
2300-441.6=1,858.40
The ordering and transportation cost C for the c mponents used in manufacturing a product is c= 100(200/x^2+ x/x+30), x≥1 where C is measured in thousands of dollars and x is the order size in hundreds. Find the rate of change of C with respect to x when (a) x = 10, (b) x = 15, and (c) x = 20. What do these rates of change imply about increasing order size?
The ordering and transportation cost C for the components used in manufacturing a product is c= 100(200/x^2+ x/x+30), x≥1 where C is measured in thousands of dollars and x is the order size in hundreds.
To find the rate of change of C with respect to x when (a) x = 10, (b) x = 15, and (c) x = 20.
The rate of change of the cost with respect to x is its derivative.
The differentiation rules to apply are
Quotient Rule: [tex]\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f^{'}(x)-f(x)g^{'}(x)}{\left[g(x)]^{2} }[/tex]
Constant Multiple Rule: [tex]\frac{d}{dx} \left[cf(x) \right] = cf^{'}(x)[/tex]
Power Rule: [tex]\frac{d}{dx} \left[x^{n} \right] = nx^{n-1}[/tex]
Constant Rule: [tex]\frac{d}{dx} [c] = 0[/tex]
So,
[tex]c^{'}(x) = 100 (\frac{x^{2}(200)^{'}-200(x^{2})^{'}}{(x^{2})^{2} }+\frac{(x+30)(x)^{'}-x(x+30)^{'} }{(x+30)^{2} } )[/tex]
[tex]c^{'}(x) = 100 (\frac{x^{2}(0)-200(2x) }{x^{4} } + \frac{(x+30)(1)-x(1+0)}{(x+30)^{2} })[/tex]
[tex]c^{'}(x) = 100 (\frac{0-400x}{x^{4} } + \frac{x+30-x}{(x+30)^{2} })[/tex]
[tex]c^{'}(x) = 100 (\frac{-400x}{x^{4} } + \frac{30}{(x+30)^{2} })[/tex]
From the above values it appears that the rate of change of the cost increases as the size of the order increases
[tex]a) c^{'}(10) = -38.13\\ \\b) c^{'}(15) = -10.37\\ \\c) c^{'}(20) = -3.8[/tex]
Hence the above values it appears that the rate of change of the cost increases as the size of the order increases
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Let sets A and B be defined as follows:
A= {j, k, l, m, q, r}
B is the set of integers greater than -2 and less than 2
(a) Find the cardinalities of A and B.
n(A) =
n(B)=
(b) Select true or false.
(I've attached a picture below to help other)
Step-by-step explanation:
(a)
it just means how many members are in a set.
there is no mystery or strange mathematical approach. we simply need to count them.
when there is an interval given, sure, we can calculate the length. but for a short interval counting is still faster.
n(A) = 6
n(B) = 3 {-1, 0, 1}
the calculation would be 2 - -2 - 1 = 4 - 1 = 3.
the "-1" comes from the fact that the end of the interval is not included.
(b)
z is not in A. as we can see : true
l is in A. as we can see again : yes. true.
-1 is in B. as I listed above : yes, true.
7 is in B. no, definitely not. false.
What does n1 mean in statistics?
The meaning of the symbol 'n,' represents the total number of individuals or observations in the sample.
What is Statistics?Statistics is the science that studies and develops methods for collecting, analyzing, interpreting, and presenting empirical data. Statistics is the study and manipulation of data, which includes methods for gathering, reviewing,
analyzing, and drawing conclusions from data. Statistics is divided into two categories: descriptive statistics and inferential statistics.
The primary goal of using statistics is to plan the data collected in terms of experimental designs and statistical surveys. Statistics is a mathematical science concerned with numerical data. In short, statistics is an important process that assists in making data-driven decisions.
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A total of 96 pints of blood were collected at the blood drive. How many quarts were collected?
Answer: 48 qt
Step-by-step explanation:
The conversion factor between pints and quarts is two which means there are 2 pints in every quart. divide the number of pints by two to find the number of quarts.
96/2 = 48
EX: 24 pints = 12 quarts
24/2 =12
EX 8 quarts = 16 pints
The same rule works in reverse but instead of dividing you multiply.
8x2 = 16
Kendra earns $11.60 per hour working at the movie theater. Each week, she donates
1
10
of her earnings to Best Friends Animal Society, her favorite charity. If Kendra worked 8
1
2
hours
Answer:
Kendra will donate $9.86 to her Best Friend's Animal Society.
Step-by-step explanation:
In order to find this answer, we need to multiply $11.60 by the 8.5 hours:
11.60 x 8.5 = $98.60
Now, if she donates 1/10 every week, we can turn 1/10 into 0.10. And on a calculator set up this equation:
0.10 x 98.60 = $9.86
May I have Brainliest please? My next rank will be the highest one: A GENIUS! Please help me on this journey to become top of the ranks! I would really appreciate it, and it would make my day! Thank you so much, and have a wonderful rest of your day!
A total of 12 waffles can be made from each batch of waffle mix. Cynthia is making 6 batches for a breakfast board meeting. If the recipe requires 5 cups of milk for each batch, how 12 many cups of milk does she need to make 6 batches of waffle mix?
2 1/2 cups of milk does she need to make 6 batches of waffle mix.
Given:
A total of 12 waffles can be made from each batch of waffle mix.
Cynthia is making 6 batches for a breakfast board meeting. If the recipe requires 5/12 cups of milk for each batch.
Number of cups of milk required = number of batches*milk required for each batch.
= 6 * 5/12
= 30/12
= 15/6
= 5/2
= 2 1/2 cups.
Therefore 2 1/2 cups of milk does she need to make 6 batches of waffle mix.
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A computer manufacturer built a new facility for assembling computers. There were construction and new equipment costs. The company paid for these costs and made combined profits of $80 million after 4 years, as shown in the graph:
There is a graph of a line with Years on the x axis and Profits in millions on the y axis that passes through 1, negative 100 and 4, 80.
Which equation most closely represents the line depicted in the graph?
a. f(x) = 60x − 160
b. f(x) = −60x + 160
c. f(x) = −80x + 4
d. f(x) = 80x − 4
The equation most closely represents the line depicted in the graph is
f(x) = 60x − 160
What is linear equation?An equation between two variables that gives a straight line when plotted on a graph.
Given a point (4,80) and a slope, m = 60, these can be used to formulate the equation of the graph. Using the point-slope formula:
y - y1 = m(x - x1)
y - 80 = 60(x - 4)
y - 80 = 60x - 240
y = 60x -160
Taking the x and y-intercepts:
At y = 0
0 = 60x - 160
x = 2.7
At x = 0
y = 60(0) - 160
y = -160
Hence, you could state that there is a graph of a line that has a y intercept that is approximately negative $160 million and an x intercept that is approximately 2.7 years.
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differentiate [tex] \sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } } [/tex]with respect to x
Step-by-step explanation:
Let [tex]{ \green{ \tt{y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4)}{3 {x}^{2} + 4x + 5}}}}} [/tex]
Take log on both sides
[tex]{ \purple{ \sf{ log_{e}y = log_{e} [\frac{(x - 3)( {x}^{2} + 4}{3 {x}^{2} + 4x + 5}]^{ \frac{1}{2}}}} } { \to}{ \tt{ {eq}^{n} (1)}}[/tex]
This above expression is in the form of
[tex]{ \boxed{ \red {\sf{ {x}^{n} = nx}}}}[/tex]
[tex]{ \boxed{ \red{ \sf{ log( \frac{m}{n} ) = log \: m \: - log \: n}}}}[/tex]
Let's apply these two formulas to eqⁿ (1) then,
[tex]{ \purple{ \sf{ log_{e}y = \frac{1}{2}[log(x - 3) + log( {x}^{2} + 4) - log( {3x}^{2} + 4x + 5) ]}}}[/tex]
differentiate with respect to x.
[tex]{ \blue{ \sf{ \frac{1}{y} \frac{dy}{dx} = \frac{1}{2}[ \frac{1}{x - 3} (1 - 0) + \frac{1}{ {x}^{2} + 4 } (2x - 0) - \frac{1}{3 {x}^{2} - 4x + 5} (6x + 4 + 0)]}}}[/tex]
[tex]{ \blue{ \sf{ \frac{dy}{dx} = y[ \frac{1}{2(x - 3)} + \frac{x}{ {x}^{2} + 4} - \frac{3x + 2}{ {3x}^{2} + 4x + 5 } ]}}}[/tex]
where [tex]{ \boxed{ \purple{ \sf{y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{3 {x}^{2} + 4x + 5}}}}} } [/tex]
Answer:
[tex]let \: \: y = \sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } } \\ \\ y =( {{\frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } }})^{ \frac{1}{2} } \\ \\ taking \: \: logarathim \ \: both \: \: sides \: \: \\ \: we \: \:get ⇒ \: ln(y) = {\frac{1}{2} ln(\frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } }) \\ \\ ⇒2 ln(y) = ln(x - 3) + ln( {x}^{2} + 4) - ln(3 {x}^{2} ) + ln(4x) + ln(5) \\ differeciate \: \: both \: \: the \: \: sides \: \: we \: \: get \: \: \\ ⇒ 2 \times \frac{1}{y} (\frac{dy}{dx} ) = \frac{1}{x - 3} + \frac{1}{ {x}^{2} + 4} .2x - \frac{1}{3 {x}^{2} } .6x + \frac{1}{4x} .4 + 0 \\ \\ ⇒ \frac{2}{y} \times \frac{dy}{dx} = \frac{1}{x - 3} + \frac{2x}{ {x}^{2} + 4} - \frac{2}{x} + \frac{1}{x} \\ \\ ⇒ \frac{dy}{dx} = \frac{y}{2} . (\frac{1}{x - 3} + \frac{2x}{ {x }^{2} + 4} - \frac{1}{x} ) \\ \\ put \: \: the \: \: value \: \: of \: \: y \: \: we \: \: get \: \\ ⇒ \frac{dy}{dx} = \frac{\sqrt{ \frac{(x - 3)( {x}^{2} + 4) }{ {3x}^{2} + 4x + 5 } } \\ }{2} \times (\frac{1}{x - 3} + \frac{2x}{ {x }^{2} + 4} - \frac{1}{x} )[/tex]
Step-by-step explanation:
used formula
[tex] { ln(m) }^{n} = n \: ln \: m \\ \\ ln(mn) = ln(m) + ln(n) \\ \\ ln( \frac{m}{n} ) = ln(m) - ln(n) \\ \\ \frac{d}{dx} ( ln(x) ) = \frac{1}{x} [/tex]
Given: a ║ b and m<6 is 1/8 of m<4, find m<4 and m<6
Answer:
∠ 4 = 160° , ∠ 6 = 20°
Step-by-step explanation:
∠ 4 and ∠ 6 are same side interior angles and sum to 180°
given ∠ 6 = [tex]\frac{1}{8}[/tex] ∠ 4 ( multiply both sides by 8 to clear the fraction )
8 ∠ 6 = ∠ 4
then
∠ 4 + ∠ 6 = 180° , that is
8 ∠ 6 + ∠ 6 = 180°
9 ∠ 6 = 180° ( divide both sides by 9 )
∠ 6 = 20°
∠ 4 = 8 × ∠ 6 = 8 × 20° = 160°
