The value of integral is 2 * arc sin((x - 2) / 2) + (x - 2) * √(4 - (x - 2)²) / 2 + C.
To solve the integral ∫ √(4x - x²) dx, follow these steps:
Complete the Square: Rewrite the quadratic expression under the square root. Start with: 4x - x²
Rearrange it to the standard quadratic form: -x² + 4x. To complete the square, factor out the negative sign: (x² - 4x)
Next, complete the square inside the parentheses:
x² - 4x = (x - 2)² - 4
So, we get:
(x² - 4x) = - [(x - 2)² - 4] = 4 - (x - 2)²
Therefore:
√(4x - x²) = √(4 - (x - 2)²)
Substitute Variables: Let u = x - 2. Then, du = dx, and the integral becomes:
∫ √(4 - u²) du
Apply the Trigonometric Substitution: Use the trigonometric substitution u = 2 sin θ. Then du = 2 cos θ dθ. Substitute these into the integral:
∫ √(4 - (2 sin θ)²) * 2 cos θ dθ
= ∫ √(4 - 4 sin² θ) * 2 cos θ dθ
= ∫ √(4 (1 - sin² θ)) * 2 cos θ dθ
= ∫ 2 cos θ * 2 cos θ dθ
= ∫ 4 cos² θ dθ
Use Trigonometric Identity: Apply the double-angle identity cos² θ = (1 + cos 2θ) / 2:
∫ 4 cos² θ dθ
= ∫ 4 * (1 + cos 2θ) / 2 dθ
= ∫ (4 + 4 cos 2θ) / 2 dθ
= ∫ (2 + 2 cos 2θ) dθ
Integrate each term:
∫ 2 dθ + ∫ 2 cos 2θ dθ = 2θ + sin 2θ + C
Back-substitute: Replace θ with the original variable x. Recall u = 2 sin θ, so θ = arcsin(u / 2). Since u = x - 2, substitute back:
θ = arcsin((x - 2) / 2)
Therefore:
2 * arcsin((x - 2) / 2) + sin(2 * arcsin((x - 2) / 2)) + C
Using the identity sin(2θ) = 2 sin θ cos θ:
sin(2 * arcsin((x - 2) / 2))
= 2 * (x - 2) / 2 * √(1 - ((x - 2) / 2)²)
= (x - 2) * √(4 - (x - 2)²) / 2
A secant is a line or segment that passes through a circle in one and only one place. True or false?
Answer:
false
Step-by-step explanation:
Can someone answer the questions on the sheet for me please?
a ball bounces from a height of 2 metres and returns to 80% of its previous height on each bounce. find the total distance travelled by the ball until it stops bouncing a ball bounces from a height of 2 metres and returns to 80% of its previous height on each bounce. find the total distance travelled by the ball until it stops bouncing
The distance traveled by the ball until it stops bouncing is:
18 meters
Step-by-step explanation:It is given that:
a ball bounces from a height of 2 meters and returns to 80% of its previous height on each bounce.
This means that the distance traveled by the ball is the distance it travels by going down as well as coming up on bouncing.
and is given as follows:
[tex]\text{Total Distance}=2+0.8\times 2\ up+0.8\times 2\ down+(0.8)^2\times 2\ up+(0.8)^2\times 2\ down+.....\\\\\text{Total distance}=2+4\times (0.8)+4\times (0.8)^2+4\times (0.8)^3+....\\\\\text{Total distance}=2+4\times [0.8+(0.8)^2+(0.8)^3+......]\\\\\text{Total distance}=2+4\times (\dfrac{0.8}{1-0.8})[/tex]
Since, the formula of infinite geometric progression is:
[tex]\sum_{n=1}^{\infty} ar^{n-1}=\dfrac{a}{1-r}[/tex]
i.e.
[tex]\text{Total distance}=2+4\times (\dfrac{0.8}{0.2})\\\\\text{Total distance}=2+4\times 4\\\\\text{Total distance}=2+16\\\\\text{Total distance}=18\ \text{meters}[/tex]
simplify completely 5x^2+9x-2/x^2+12x+20*x^2+17x+70/15x-3 ...?
In 0.57 which is the part and which is the whole
10 is %20 of what number
Answer:
50
Step-by-step explanation:
10=20/100*50/1=1000/100=10
Statistics show that the sales force of Golden Wholesalers successfully closed 1,711 sales out of 1,950 sales calls. What was their percent success rate?
Chandra has 2 liters of a 14% solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 7 liters of a 34% solution?
A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats . How many seats are in thecivic theater
There are 840 seats in total in Cinema Hall.
What is arithmetic Progression?The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence.
Given:
First term, [tex]a_1[/tex] = 22
second row = 22 + 3 = 25
Third row = 25 + 3 = 28
So, the Recursive Formula is
[tex]a_n[/tex] = a + (n-1)d
[tex]a_n[/tex] = 10 + (n-1)3
[tex]a_n[/tex] = 10 + 3n - 3
[tex]a_n[/tex] = 7 + 3n
In 21th rows the seats are
= 7 +3 (21)
= 7 + 63 = 70
Now, to find the number of seats in hall we have to find the sum of seats from row 1 to row 21 using Formula
= n×([tex]a_1[/tex]+ [tex]a_n[/tex])/2
= 21 x (10 + 70) /2
= 21 x 80 /2
= 21 x 40
= 840
Learn more about arithmetic Sequence here:
https://brainly.com/question/29854857
#SPJ6
A triangle has sides of square root of 2 and 3. Which could not be the length of the third side if it is a right triangle?
Answer:
the root of 13
Step-by-step explanation:
What percent of 150 is 135?
15-28i=3l+(4m)i solve ...?
To solve the equation 15-28i=3l+(4m)i, separate the real and imaginary parts of the equation and solve for l and m separately.
Explanation:To solve the equation 15-28i=3l+(4m)i, we can separate the real and imaginary parts of the equation. The real part is 15, and the imaginary part is -28. Similarly, the real part of the right side is 3l and the imaginary part is 4m. Equating the real parts and imaginary parts separately, we get two equations: 15 = 3l and -28 = 4m. Solving these equations will give us the values of l and m.
In the first equation, dividing both sides by 3 gives us l = 5. In the second equation, dividing both sides by 4 gives us m = -7.
Therefore, the solution to the equation 15-28i=3l+(4m)i is l = 5 and m = -7.
Find the numerical value of cosh (ln5)
The numerical value of cosh(ln5) is 2.55, obtained by using the hyperbolic cosine function definition and the properties of exponential and logarithmic functions.
Explanation:The question asks you to find the numerical value of cosh (ln5). This involves using hyperbolic trigonometry functions, specifically the hyperbolic cosine function.
The hyperbolic cosine, represented by 'cosh', is a mathematical function whose definition is similar to the ordinary trigonometric cosine function. However, it is defined using exponential functions rather than angular measure. The general definition is cosh(x) = (e^x + e^(-x))/2.
Using this formula and substituting ln5 for x results in cosh(ln5) = (e^(ln5) + e^(-ln5))/2.
The expression e^(ln5) simplifies to 5 since e and ln are inverse functions.
For the second term, e^(-ln5), we can use the fact that a negative exponent signifies taking the reciprocal, so this simplifies to 1/5.
Therefore, cosh(ln5) = (5 + 1/5)/2 = 5.1/2 = 2.55.
Learn more about Hyperbolic functions here:https://brainly.com/question/32479160
#SPJ12
The solution to cosh (ln5) is obtained by substituting ln5 in the formula of hyperbolic cosine function, which simplifies it to (5 + 1/5)/2 yielding a result of 2.6.
To find the numerical value of cosh (ln5), we need to know the definition of cosh. It is a hyperbolic function defined as cosh(x) = (e^x + e^-x)/2 where e is Euler's number (approximately 2.71828).
So, to find cosh (ln5), we substitute ln5 in place of x in the formula cosh(x) = (e^x + e^-x)/2.
This gives us cosh(ln5) = (e^ln5 + e^-ln5)/2. However, e and ln are inverse functions, so e^ln5 simplifies to 5. Similarly, for e^-ln5, we need to use the property a^-b = 1/a^b to simplify it to 1/5.
Then we just add and divide, yielding cosh(ln5) = (5 + 1/5)/2 = 2.6.
Learn more about hyperbolic cosine function here:
https://brainly.com/question/2254116
#SPJ6
A sail is in the form of a right triangle that is four times as high as it is wide.The sail is made from 32 square meters of material. What is its height?
To calculate the height of a sail that is four times as high as it is wide and made from 32 square meters of material, we use the area formula for a triangle and find that the height is 16 meters.
The student's question concerns the height of a sail in the shape of a right triangle, where the sail's height is four times its width and the total area of the sail is 32 square meters. To find the height, we can let the width be x meters, which makes the height 4x meters due to the given ratio.
Using the formula for the area of a triangle, A = bh/2, where b is the base and h is the height, we can write an equation for the area in terms of x: 32 m² = x × (4x) / 2.
Simplifying this, we get 32 = 2x2, which can be further simplified to x² = 16. Taking the square root of both sides gives us x = 4. Therefore, the height of the sail, being four times the width, is 16 meters.
Find the value of x for which p is parallel to q , if m<1=(3x) and m<3=105. The diagram is not scale.
Find a value of the standard normal random variable Z, Call it Z0, such that
a. P(Z ≤ Z0) = .0401
b. P(- Z0 ≤ Z ≤ Z0) = .95
c. P(- Z0 ≤ Z ≤ Z0) = .90
d. P(- Z0 ≤ Z ≤ 0) = .2967
The 0 is subzero, don't know why the site couldn't show it.
Using the normal distribution, it is found that:
a) [tex]Z_0 = -1.75[/tex].
b) [tex]Z_0 = 1.96[/tex].
c) [tex]Z_0 = 1.645[/tex].
d) [tex]Z_0 = -0.83[/tex].
In a normal distribution 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]
Each z-score has a p-value associated with it, which is the percentile of X, and is found at the z-table.Item a:
This is Z with a p-value of 0.0401, thus [tex]Z_0 = -1.75[/tex].
Item b:
Due to the symmetry of the normal distribution, the middle 95% is between the 2.5th and the 97.5th percentile.We want the positive value, so Z with a p-value of 0.975, which is [tex]Z_0 = 1.96[/tex]Item c:
Same logic as b, just middle 90%, thus [tex]Z_0 = 1.645[/tex].
Item d:
This is Z with a p-value of 1 - 0.2967 = 0.2033, thus [tex]Z_0 = -0.83[/tex].
A similar problem is given at https://brainly.com/question/12982818
Final answer:
The value of the standard normal random variable Z that satisfies the equation P(-Z0 ≤ Z ≤ Z0) = 0.90 is approximately 1.28.
Explanation:
The z-score that corresponds to the area 0.90 under the standard normal distribution can be found using a z-table or a calculator.
Using a z-table, locate the area 0.90. The corresponding z-score is approximately 1.28.Using a calculator, the invNorm(0.90) command can be used to find the z-score, which is also approximately 1.28.Therefore, the value of the standard normal random variable Z, denoted as Z0, that satisfies the equation P(-Z0 ≤ Z ≤ Z0) = 0.90 is approximately 1.28.
Suppose you buy a car with a value of $9,250. Each year the value of your car will depreciate by 5.1%. How much will your car be worth in 8 years?
Final answer:
To calculate the value of the car in 8 years with a constant yearly depreciation rate of 5.1%, we can use the formula for compound interest.
Explanation:
To calculate the value of the car in 8 years, we can use the formula for compound interest: A = P(1 - r/n)^(nt), where A is the final amount, P is the initial amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.
In this case, the initial amount is $9,250, the interest rate is 5.1%, and the number of times interest is compounded per year is 1. Plugging these values into the formula, we get:
A = 9250(1 - 0.051/1)^(1*8) = $6630.27
Therefore, the car will be worth approximately $6,630.27 in 8 years.
Pre-Calc :
How do you get the direct relationship between y and x and how do you know if the parametric equations determine y as a function of x with the following?
x = 2t and y = 3t -1
9x^2-y^2=1
(a) find y' by implicit differentiation
(b) Solve the equation explicitly for y and differentiate to get y' in terms of x
(c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a)
Finding the rate of change in something with respect to something is known as differentiation. For Example, Change in y with respect to x is known as the derivative of y with respect to x.
How to solve it?(a) [tex]9{x}^2-{y}^2 = 1[/tex]
Differentiating equation with respect to x :
[tex]9*2x - 2y\frac{dy}{dx} = 0\\ 18x = 2y\frac{dy}{dx}\\9\frac{x}{y} =\frac{dy}{dx}[/tex]
(b)
[tex]9{x}^2-{y}^2 = 1\\9{x}^2-1 = {y}^2 \\\sqrt[2]{9{x}^2-1} = y\\[/tex]
differentiating with respect to x gives:
[tex]\frac{1}{2\sqrt{9{x}^2-1} } *(18x) = \frac{dy}{dx}\\\frac{1}{\sqrt{9{x}^2-1} } *(9x) = \frac{dy}{dx}[/tex]
(c) [tex]9\frac{x}{y} =\frac{dy}{dx}[/tex]
substituting value of y
[tex]\sqrt[2]{9{x}^2-1} = y\\\\\frac{9x}{\sqrt[2]{9{x}^2-1}} = \frac{dy}{dx}[/tex]
Hence the solution is consistent.
To learn more about Differentiation visit:
https://brainly.com/question/2321397
#SPJ2
40% of the students at Rockledge Middle School are musicians. 75% of those musicians have to read sheet music when they play their instruments If 38 of the students can play their instruments without reading sheet music, how many students are there at Rockledge Middle School?
What is the simplified form of the expression? 7^4
a. 16,384
b. 16,807
c. 343
d. 2,401
Answer:
The correct answer is D. The simplified form of the expression 7^4 is 2,401.
Step-by-step explanation:
7 ^ 4, that is, the exponentiation of 7 to the fourth power, implies the continuous multiplication of the number 7 for 4 times. Therefore, this number will appear 4 times in a multiplication, whose result will be said number exposed to the fourth power.
Simplified, this operation translates into 7x7x7x7, whose result is 2,401.
640, 160, 40, 10, ...
Which correctly describe the graph of the geometric sequence? Check all that apply.
The graph will show exponential growth.
The graph will appear linear.
The domain will be the set of natural numbers.
The range will be the set of natural numbers.
The graph will show exponential decay.
The graph will show exponential decay and The domain will be the set of natural numbers.
Option C and D will apply.
What is a Geometric Sequence ?A geometric sequence is a sequence of numbers obtained by either multiplying or dividing by the same value.
The geometric sequence given is
640, 160, 40, 10, ...
Each number is divided by 4
So the nth term will be
[tex]\rm T_n = a_1 r^{n-1}[/tex]
where [tex]\rm a_1 \; is \; the \; first \; term[/tex]
and r is the common ratio
r =(1/4) for this sequence
The function is
[tex]\rm T_n = 640(\dfrac {1}{4})^{n-1}[/tex]
As from the function it can be understood , That The graph will show exponential decay and The domain will be the set of natural numbers.
Option C and D will apply.
To know more about Geometric Sequence
https://brainly.com/question/11266123
#SPJ5
The sequence is a geometric sequence showing exponential decay with a factor of 0.25 (divisible by 4). The graph will show exponential decay and not appear linear. The domain is the set of natural numbers, while the range consists of specific values from the sequence, not all natural numbers.
Explanation:The sequence given is 640, 160, 40, 10, ..., which is a geometric sequence. We can determine this because each term is obtained by multiplying the previous term by a constant factor.
In this case, each term is divided by 4 to get the next term (160 is 640 divided by 4, 40 is 160 divided by 4, and so on). Therefore, the common ratio r is ¼ or 0.25, which is less than 1.
Now, let’s evaluate the statements provided:
The graph will show exponential decay because the ratio is between 0 and 1, which means the sequence is getting smaller.The graph will not show exponential growth since the terms are decreasing. Exponential growth would require the terms to get larger as the sequence progresses.The graph will not appear linear because the rate of decrease is not constant in terms of subtraction, but rather in division by a factor.The domain of a sequence typically consists of the set of natural numbers since it represents the position of each term in the sequence.The range will not be the set of natural numbers because the terms do not include all natural numbers but rather specific values determined by the geometric sequence.So, the correct statements are that the graph will show exponential decay and the domain will be the set of natural numbers.
Learn more about Geometric Sequence here:https://brainly.com/question/34721734
#SPJ3
Andrea drove 500 miles in 10 hours. find the average number of miles per hour that andrea drove
find g'(4) given that f(4)=3 and f'(4) = -5. g(x) = f(x)/x
Jessica's family is on vacation at the Outer Banks. She is going to buy dinner at a local seafood distributor. She can buy shrimp at $8.50 per pound and crab cakes for $3 each. she has $45 to spend and must buy 3 pound of shrimp. What is the largest quantity of crab cakes that she can buy?
Rewrite y = √25-75) + 3 to make it easy to graph using a translation. Describe the graph
A.Rewritten, y = √X-3 + 3. It is the graph of y = √X translated 3 units right and 3 units up.
B. Rewritten, y = √X-3 + 3. It is the graph of y = √X translated 3 units left and 3 units up.
C. Rewritten, y = √X-3 + 3. It is the graph of y = √X translated 3 units left and 3 units up.
D. Rewritten, y = √X-3 + 3. It is the graph of y = √X translated 3 units right and 3 units up.
2. How is the graph of Y=√X) -5 translated from the graph of √X ?
shifted 5 units right
shifted 5 units down
shifted 5 units left
shifted 5 units up
50 coins is worth $5.20. There areally 12 more nickels and dimes, and the rest Andre quarters. How many coins of each type are there?
find the median for the given set of data.
0.6, 1.1, 0.87, 2, 0.87, 1.23
a) 0.87
b)0.985
c)1.1
The LCM of 50, 120, and 225 is 5. true or false?
Answer:
Its false
Step-by-step explanation:
well first i pt the answer in and it came out correct.
I HAVE 5 MATH QUESTIONS AND NEED HELP ASAP! ONLY COMMENT IF YOU ARE GOING TO HELP.