The limit of csc(3x)/cot(x) as x approaches 0 is 1, after applying trigonometric identities and the properties of limits.
Explanation:The student is asking for the limit of the function csc(3x)/cot(x) as x approaches 0. This is a limit problem in calculus, which is a part of mathematical analysis that deals with the behavior of functions near certain points. To find the limit as x approaches 0 for the function csc(3x)/cot(x), we need to apply trigonometric identities and the properties of limits. The function csc(3x) can be rewritten as 1/sin(3x) and cot(x) as cos(x)/sin(x). So the original limit becomes:
lim(x → 0) [csc(3x)/cot(x)] = lim(x → 0) [(1/sin(3x)) × (sin(x)/cos(x))].
Using the trigonometric limit lim(x → 0) (sin(x)/x) = 1 and lim(x → 0) (cos(x)) = 1, we can simplify the expression by substituting 3x for x in the limit sin(x)/x. We get:
lim(x → 0) [(1/sin(3x)) × (sin(x)/cos(x))] = lim(x → 0) [(sin(x)/cos(x)) × (1/(3sin(3x)))] × (3x/x).
As x approaches 0, cos(x) approaches 1 and 3x/x approaches 3, hence:
lim(x → 0) [(sin(x)/cos(x)) × (1/(3sin(3x)))] × (3x/x) = lim(x → 0) (sin(x)/x) × lim(x → 0) 1/cos(x) × lim(x → 0) (3x/3sin(3x)) = 1 × 1 × (3/3) = 1.
The final limit of csc(3x)/cot(x) as x approaches 0 is 1.
How do you Solve by factoring
Final answer:
To solve an equation by factoring, find the factors, set them equal to zero, and solve for the variable.
Explanation:
To solve an equation by factoring, you need to find the factors of the equation and set each factor equal to zero. Then, solve each equation separately to find the values of the unknown variable. Let's take an example:
If we have the equation [tex]x^2 - 5x + 6 = 0[/tex]:
Factor the quadratic expression: (x - 2)(x - 3) = 0Set each factor equal to zero and solve: x - 2 = 0 and x - 3 = 0solve each equation separately to find the values of x: x = 2 and x = 3This method of factoring and setting each factor equal to zero is a systematic approach to solving quadratic equations, providing a structured way to identify the solutions for the unknown variable. The application of this technique is fundamental in algebra, enabling the efficient determination of roots in quadratic expressions and contributing to a comprehensive understanding of solving equations.
For what values of x is the following inequality true? 5/7 + x/7 >(underlined) 10. PLZ HELP!!! 12 points
4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
(x+5) - (10 * 7) / 7 = x - 65 / 7
Equation at the end of step 4:
x - 65 / 7 > 0
Step 5: 5.1 Multiply both sides by 7
Solve Basic Inequality:
Find the value of y.
–2y + 16 + 6y = 44
a. 15
b. 7
c. 3.5
d. –6
If dh = 3x-3 and fh=x+7 find the value of x for which defg must be a parallelogram
parallelogram
DH = FH
3x -3 = x + 7
2x = 10
x = 5
Answer
c. 5One day, a person went to a horse racing area. instead of counting the number of humans and horses, he counted 72 heads and 192 legs. how many humans and horses were there?
State how many imaginary and real zeros the function has. f(x) = x3 + 5x2 - 28x - 32
All equation has always
even numbers of imaginary roots because they always come in pairs. Therefore
there must be either 2 imaginary roots, or zero.
To determine the types of roots, we could try synthetic
division by some of the factors of the number -32. The simplest factors are +1
or -1. Using this, we can quickly show that (x + 1) is a factor, and leaving
(x² + 4x - 32) as the remainder. This remainder equation then factorises to (x
- 4) (x + 8).
So to conclude there are three real zeros which are:
x = -1
x = 4
x = -8
simplify
3√45
a)5√15
b)9√5
c)15√9
A company has found that its supply function is equal to the square of the price of their product, all divided by three. The market for its products is also related to price. demand is a basic 920 units minus thirty times price, minus a quarter of the square of the price. What is the ideal price of their product, to make sure that there is neither a surplus nor a shortage? How many units will they sell at this price?
How many different three digit locker combinations are possible if the digits are able to repeat?
A.30
B.1000
C.300
There are 1000 three digit locker combinations are possible if the digits are able to repeat.
The correct option is B.
If the digits are able to repeat, there are 10 options for each digit (0-9) in a three-digit combination.
Since each digit can be chosen independently, the total number of different three-digit combinations is given by:
Number of combinations
= Number of options for the first digit x Number of options for the second digit x Number of options for the third digit
Number of combinations = 10 x 10 x 10 = 1000
Learn more about Combination here:
https://brainly.com/question/13058214
#SPJ4
if 16 + 4x is 10 more than 14, what is the value of 8x?
The sum of two numbers is 41 and the difference is 19 . what are the numbers?
A plane is defined by two lines. What's the counterexample
For her phone service, Lucy pays a monthly fee of $18, and she pays an additional $0.05 per minute of use. The least she has been charged in a month is $81.20What are the possible numbers of minutes she has used her phone in a month? Use m for the number of minutes and solve your inequality for m.
A recipe for guacamole uses 12% lime juice. if a batch contains 0.75 cup of kime juice, how large is the batch of guacamole
What does the y intercept mean
solve
4x + 6 = 38
A. 144
B. 11
C. 8
D. 3.5
IM USING UP SO MANY POINTS CAUSE PEOPLE WONT ANSWER PLZ HELP-A business owes $8000 on a loan. Every month, the business pays 12 of the amount remaining on the loan.
How much will the business pay in the sixth month?
What is the solution to 2h+8>3h-6
What is the answer to H(x)=8x-10
John's gas tank is 16 full. after he buys 5 gallons of gas, it is 12 full. how many gallons can john's tank hold?
Coach Stevens needs to purchase sprinklers to water the baseball field. The standard distance between bases is 90 feet and the infield is a perfect square.
Coach Stevens found one sprinkler that sprays a maximum distance of 50 feet.
If Coach Stevens placed the sprinkler on the pitcher's mound, would the water spray so that the entire infield was watered? Justify your response.
Answer:
So, if coach Stevens places as sprinkler of maximum distance 50 ft, he won't be able to cover the entire infield.
Step-by-step explanation:
Consider the infield as a polygon ABCDE.
It is given that each side AB, BC, CD, and DA is of length 90ft. It is also given that the infield is a perfect square.
Thus, the area of the square ABCD = 8100 ft.
Now consider the triangle BCD. Because of symmetry, the area of this triangle equals half the area of the square. Again, because of symmetry, the length of the base BD equals twice the length of the height EC.
Thus from to the formula of area of the triangle:
base × height = 4050 base × height = 8100 2(height) × height= 8100 height63.64 ft
What is the 4 digit number in which the first digit is one-fifth of the last, and the second and third digits are the last digit multiplied by 3?
Jerry has 64 cupcakes. Jane has 5 more cupcakes. How many cupcakes does Jane and Jerry have together?
in a 30-60-90 Blondie triangle the long leg is half the hypotenuse
A. always
B. sometimes
c never
Answer:
never
Step-by-step explanation:
In a 30 - 60 - 90 degree triangle , the common ratio is
1 : [tex]\sqrt{3}[/tex] : 2
Shorter leg that is opposite to 30 degree = 1
Longer leg that is opposite to 60 degree = [tex]\sqrt{3}[/tex]
Hypotenuse is opposite to 90 degree = 2
Shorter leg is half the length of the hypotenuse as per the ratio
Long leg is never the half of the hypotenuse
1. Main Show Tank Calculation:
The main show tank has a radius of 60 feet and forms a quarter sphere where the bottom of the pool is spherical and the top of the pool is flat. (Imagine cutting a sphere in half vertically and then cutting it in half horizontally.) What is the volume of the quarter-sphere shaped tank? Round your answer to the nearest whole number.
2. Holding Tank Calculations:
The holding tanks are congruent. Each is in the shape of a cylinder that has been cut in half vertically. The bottom of each tank is a curved surface and the top of the pool is a flat surface. What is the volume of both tanks if the radius of tank #1 is 30 feet and the height of tank #2 is 110 feet?
3. The company is building a scale model of the theater’s main show tank for an investor's presentation. Each dimension will be made ⅛ of the original dimension to accommodate the mock-up in the presentation room. What is the volume of the smaller mock-up tank?
4. Using the information from #4, answer the following question by filling in the blank: The volume of the original main show tank is ____% of the mock-up of the tank.
Which shape has the same volume as the given rectangular prism?
Thank you!
Answer:
The correct option is (A) i.e. cylinder.
Step-by-step explanation:NET BANGERZ
Which exponential function goes through the points (1, 8) and (4, 512)?
f(x) = 4(2)^x
f(x) = 8(2)^-x
f(x) = 4(4)^-x
f(x) = 2(4)^x
Answer:
Therefore, the correct option is: (C.) [tex]\( f(x) = 2 \cdot 4^x \)[/tex]
Explanation:
To determine the exponential function that passes through the given points, we can use the general form of an exponential function:
[tex]\[ f(x) = a \cdot b^x \][/tex]
Where:
[tex]\( a \)[/tex] is the initial value or the value of the function when [tex]\( x = 0 \).[/tex]
[tex]\( b \)[/tex] is the base of the exponential function.
Given the points [tex]\((1, 8)\) and \((4, 512)\)[/tex], we can substitute these coordinates into the general form and solve for [tex]\( a \) and \( b \).[/tex]
Using the point [tex]\((1, 8)\):[/tex]
[tex]\[ 8 = a \cdot b^1 \][/tex]
[tex]\[ 8 = a \cdot b \][/tex]
Using the point [tex]\((4, 512)\):[/tex]
[tex]\[ 512 = a \cdot b^4 \][/tex]
We can divide the equation obtained from the second point by the equation obtained from the first point to eliminate [tex]\( a \)[/tex] and solve for [tex]\( b \):[/tex]
[tex]\[ \frac{512}{8} = \frac{a \cdot b^4}{a \cdot b} \][/tex]
[tex]\[ 64 = b^3 \][/tex]
Taking the cube root of both sides:
[tex]\[ b = 4 \][/tex]
Now that we have found the value of [tex]\( b \)[/tex], we can substitute it back into one of the equations to find \( a \). Let's use the first point [tex]\((1, 8)\):[/tex]
[tex]\[ 8 = a \cdot 4 \][/tex]
[tex]\[ a = 2 \][/tex]
So, the exponential function that passes through the points [tex]\((1, 8)\) and \((4, 512)\)[/tex] is:
[tex]\[ f(x) = 2 \cdot 4^x \][/tex]
multiply (2.1 x 10^3) x (3.5 x 10^2)
A. 7.35 x 10^5
B. 3.6 10^1
C. 6.9 x 10^3
D. 2.83 x 10^2
What is another name for m1?
Write in slope-intercept form of the equation of the line through the given point. Through: (-3,-3) and (4,0)
Answer:
y = 3/7 x - 12/7
Step-by-step explanation:
The slope is (0+3)/(4+3) = 3/7
So, now you have a point and a slope, so use the point-slope form. That gives you
y-0 = 3/7 (x-4)
Now just rearrange to slope-intercept form.
y = 3/7 x - 12/7