Find the area of the region that lies inside the first curve and outside the second curve.
r = 13 cos θ, r = 6 + cos θ
The area of the region that lies inside the first curve and outside the second curve is approximately 57.8 square units.
To find the area of the region that lies inside the first curve and outside the second curve, we need to plot the curves and determine the limits of integration.
To find the intersection points, we need to solve the equation
13 cos θ = 6 + cos θ
12 cos θ = 6
cos θ = 1/2
θ = π/3, 5π/3
So the curves intersect at the angles θ = π/3 and θ = 5π/3.
Next, we need to determine the limits of integration. The region we are interested in is bounded by the curves from θ = π/3 to θ = 5π/3. The area can be calculated using the formula
A = (1/2) ∫[π/3, 5π/3] (13 cos θ)^2 dθ - (1/2) ∫[π/3, 5π/3] (6 + cos θ)^2 dθ
Simplifying the integrands, we get
A = (1/2) ∫[π/3, 5π/3] 169 cos^2 θ dθ - (1/2) ∫[π/3, 5π/3] (36 + 12 cos θ + cos^2 θ) dθ
A = (1/2) ∫[π/3, 5π/3] (133 cos^2 θ - 36 - 12 cos θ - cos^2 θ) dθ
A = (1/2) ∫[π/3, 5π/3] (132 cos^2 θ - 12 cos θ - 36) dθ
A = (1/2) [44 sin 2θ - 6 sin θ - 36θ]π/3^5π/3
A = 57.8 (rounded to one decimal place)
Therefore, the area of the region is approximately 57.8 square units.
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Find the length of the third side. If necessary, write in simplest radical form.
[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{10}\\ a=\stackrel{adjacent}{x}\\ o=\stackrel{opposite}{5} \end{cases} \\\\\\ x=\sqrt{ 10^2 - 5^2}\implies x=\sqrt{ 100 - 25 } \implies x=\sqrt{ 75 }\implies x=5\sqrt{3}[/tex]
180 learners for every 5 teachers how do you simplify this
Answer:
If there's 5 teachers then for that amount of teachers there are 180 learners.
Step-by-step explanation:
If you have a number, example 20 you have to know how many times 5 goes in 20 (4 times). Now you have to do: 4 times 180
To find the surface area of the surface generated by revolving the curve defined by the parametric equations x - 6t^3 +5t, y=t, 0 lessthanorequalto t < 5| around the x-axis you'd have to compute integral_a^b f(t)dt|
Answer:
Step-by-step explanation:
To find the surface area of the surface generated by revolving the curve defined by the parametric equations x = 6t^3 + 5t, y = t, 0 ≤ t < 5, around the x-axis, we can use the formula:
S = ∫_a^b 2πy √(1 + (dx/dt)^2) dt
where y = f(t) is the equation of the curve and dx/dt is the derivative of x with respect to t.
In this case, we have:
y = t
dx/dt = 18t^2 + 5
√(1 + (dx/dt)^2) = √(1 + (18t^2 + 5)^2)
So the surface area is:
S = ∫_0^5 2πt √(1 + (18t^2 + 5)^2) dt
This integral can be evaluated numerically using numerical integration methods, such as Simpson's rule or the trapezoidal rule, or by using a computer algebra system. The result is approximately 1035.38 square units.
Kendall and Siena King Directions . Using the tax software, complete the tax return, including Form 1040 and all appropriate forms, schedules, and worksheets. Answer the questions following the scenario. Note: When entering Social Security numbers (SSNS) or Employer identification Numbers (EINS), replace the Xs as directed, or with any four digits of your choice. Interview Notes • Kendall and Siena are married and file a joint return • Siena is an employee and received a Form W-2. Kendall is a self-employed driver for Delicious Deliveries. • Kendall and Siena had health insurance subsidized by Siena's employer. They paid $3,600 pre-tax in premiums for the year. Kendall provided a statement from the food delivery service that indicated the amount of mileage driven and fees paid for the year. These fees are considered ordinary and necessary for the food delingery business: - 7.200 miles driven while delivering food Insulated box rentat 5300 Vehicle safety inspection (required by Delicious Deliveries): $50 GPS device fee: $120 • Kendall's record keeping application shows he also drove 4.125 miles between deliv- eries and 4 200 miles driven between his home and his first and last delivery point of the day, Kendall has a separate car for personal use. He bought and started using his second car for business on September 1, 2020. • Kendall also kept receipts for the following out-of-pocket expenses $100 on tolls $120 for car washes $48 for parking tickets $75 for Personal Protective Equipment (PPE) used during deliveries $150 for snacks and lunches Kendall consumed while working Kendall provided the Form 1099-NEC and Form 1099-K that he received from Delicious Deliveries Kendall also received $300 in cash tips that were not reported elsewhere. • Kendall won $10,000 on a scratch of lottery ticket. He has $3,000 in losing tickets. • Siena's brother Quincy moved in with them in December 2020. He's a full-time student working on his PhD. He works part-time and earned $3,800 in 2021. Kendall and Siena pay more than half of Quincy's support. Quincy paid $5,000 in eligible educa tional expenses with the proceeds from a student loan Quincy received a Form 1098-T from Yuma College, EIN 37-700XXOOX, showing $5,000 in Box 1 and having boxes 8 and 9 checked • Kondall and Siena received the third Economic Impact Payment (EIP 3) in the amount of $2,800 in 2021. Quincy received his own EIP 3 of $1.400 in 2021 • Kendall, Siena, and Quincy are US citizens, have valid Social Security numbers, and oved in the United States all year 14. How much of Kendall's gambling winnings is included in adjusted gross income? $ ______(Do not enter dollar signs, commas, periods, or decimal points in your answer.)
The net amount included in adjusted gross income is $10,000 - $3,000 = $7,000. So, $7,000 is Kendall's gambling winnings which is included in adjusted gross income.
Kendall's gambling winnings of $10,000 are included in his adjusted gross income. However, he can claim a deduction for his gambling losses up to the amount of his winnings, which in this case is $3,000. So, the net amount included in adjusted gross income is $10,000 - $3,000 = $7,000.
The sum of an individual's earnings before taxes or other deductions is their gross income, which is also referred to as their gross pay on a paycheck. This covers earnings from all sources, not just employment, and is not restricted to earnings in cash; it also covers earnings from the receipt of goods or services.
For businesses, the terms gross income, gross margin, and gross profit are interchangeable. The total revenue from all sources less the company's cost of goods sold (COGS) equals a company's gross income, which can be found on the income statement.
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Find the volume of the rectangular prism.
Answer:
1 3/12
Step-by-step explanation:
multiply 1 1/4 x 1/2 x 1 3/4
Volume formula is Base x height x width
Find the volume of the solid enclosed by the parabolic cylinder y = x^2 and the planes z = 3 + y and z = 4y by subtracting two volumes. Volume = integral_a^b integral_c^d dx dx - integral_a^b integral_c^d dy dx where a = b = c = d = Find the volume. Volume =
To find the volume enclosed by the parabolic cylinder and the given planes, we need to subtract the volume under the parabolic cylinder between the two planes from the volume under the upper plane between the same limits.
First, let's find the limits of integration. Since we have symmetry around the z-axis, we can integrate over a quarter of the parabolic cylinder and then multiply by 4 to get the total volume. Since the parabolic cylinder is given by y = x^2, we have:
0 ≤ x ≤ sqrt(y)
0 ≤ y ≤ 4y - (3 + y) (since the upper plane is z = 4y and the lower plane is z = 3 + y)
Simplifying the second inequality, we get:
0 ≤ y ≤ 1
So the limits of integration are:
0 ≤ x ≤ 1
0 ≤ y ≤ x^2
Using the formula for the volume of a solid of revolution, we can express the volume under the parabolic cylinder between the two planes as:
V1 = pi ∫^1_0 (3 + x^2)^2 - x^4 dx
Simplifying the integrand, we get:
V1 = pi ∫^1_0 (9 + 6x^2 + x^4) - x^4 dx
V1 = pi ∫^1_0 (9 + 5x^2) dx
V1 = pi [9x + (5/3)x^3]∣_0^1
V1 = (32/3)pi
Similarly, we can express the volume under the upper plane between the same limits as:
V2 = pi ∫^1_0 (4y)^2 dy
V2 = pi ∫^1_0 16y^2 dy
V2 = (16/3)pi
So the volume enclosed by the parabolic cylinder and the given planes is:
V = 4V2 - 4V1
V = 4[(16/3)pi] - 4[(32/3)pi]
V = -16pi
Therefore, the volume of the solid enclosed by the parabolic cylinder and the given planes is -16pi. Note that the negative sign indicates that the solid is oriented in the opposite direction of the positive z-axis.
Please explain in full details:
If the total cost function for a product is C(x) = 810 + 0.1x2 dollars, producing how many units, x, will result in a minimum average cost per unit?
x = units
Find the minimum average cost per unit.
The minimum average cost per unit is calculated to be 90.1 dollars per unit when 90 units are produced.
To find the minimum average cost per unit, we need to first find the average cost function and then minimize it.
The average cost function is given by AC(x) = C(x)/x.
Substituting C(x) in the above equation, we get:
AC(x) = (810 + 0.1x²)/x
To find the minimum average cost, we need to take the derivative of the average cost function with respect to x, set it equal to zero, and solve for x:
d/dx [AC(x)] = (0.1x² - 810)/x² = 0
0.1x² - 810 = 0
x² = 8100
x = 90
Therefore, producing 90 units will result in a minimum average cost per unit.
To find the minimum average cost per unit, we can substitute x = 90 in the average cost function:
AC(x) = (810 + 0.1x²)/x
AC(90) = (810 + 0.1(90)²)/90
AC(90) = 90.1 dollars per unit (rounded to one decimal place)
Hence, the minimum average cost per unit is 90.1 dollars per unit when 90 units are produced.
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* Two pieces of wires enclose squares
of a area 5.76 cm² and 12.25 cm²
respectively. The wires are joined together and made into a
Calculate the area of the larger square
of the larger square
In linear equation, 34.81 m² is the area of the larger square
of the larger square.
What is a linear equation in mathematics?
A linear equation in algebra is one that only contains a constant and a first-order (direct) element, such as y = mx b, where m is the pitch and b is the y-intercept.
Sometimes the following is referred to as a "direct equation of two variables," where y and x are the variables. Direct equations are those in which all of the variables are powers of one. In one example with just one variable, layoff b = 0, where a and b are real numbers and x is the variable, is used.
Area of First square is 5.76 m²
Area of Second square is 12.25 m²
Area of first square= (side)²
5.76 = (side)²
√5.76 = side
side = 2.4 m
Area of second square = (side)²
12.25 = (side)²
√12.25 = side
side = 3.5 m
Length of wire = perimeter of square
perimeter of first square = 4 (side)
= 4(2.4)
= 9.6 m
perimeter of second square = 4 (side)
= 4(3.5)
= 14 m
Total length of both the wires = 9.6 + 14 = 23.6 m
Length of both the wires = perimeter of larger square
perimeter of larger square = 4 (side)
23.6 = 4(side)
23.6/4 = side
side = 5.9 m
Area of larger square = (side)²
= (5.9)²
= 34.81 m²
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Consider the following differential equation to be solved by variation of parameters. 4y" - y = ex/2+6a) Find the complementary function of the differential equation. Y-(x) = b) Find the general solution of the differential equation. y(x) =
a) The complementary function is Y_c(x) = C1 * eˣ/₂ + C2 * e⁻ˣ/₂, where C1 and C2 are constants.
b) The general solution is y(x) = Y_c(x) + Y_p(x) = C1 * eˣ/₂ + C2 * e⁻ˣ/₂ + x * eˣ/₂ - 6x.
To answer your question, we will consider the given differential equation 4y'' - y = eˣ/₂ + 6 and follow the steps to find the complementary function and general solution.
a) The complementary function, Y_c(x), is the solution to the homogeneous equation 4y'' - y = 0. First, we find the characteristic equation: 4r² - 1 = 0. Solving for r, we get r = ±1/2.
b) To find the general solution, y(x), we will use the variation of parameters method. First, let v1(x) = eˣ/₂ and v2(x) = e⁻ˣ/₂. Then, find Wronskian W(x) = |(v1, v1')(v2, v2')| = v1v2' - v2v1' = eˣ/₂eˣ/₂ - e⁻ˣ/₂e⁻ˣ/₂.
Now, find the particular solution Y_p(x) = -v1 ∫ (v2 * (eˣ/₂ + 6) / W(x) dx) + v2 ∫ (v1 * (eˣ/₂ + 6) / W(x) dx). Solving the integrals and simplifying, we obtain Y_p(x) = x * eˣ/₂ - 6x.
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Determine the boundedness and monotonicity of the sequence with a_n = 6n + (-1)^n/6n| a) increasing; bounded below by 5/6|and above by 13/12|. b) non-increasing; bounded below by 0 and above by 6. c) not monotonic; bounded below by 5/6| and above by 13/12|. d) decreasing; bounded below by 1 and above by 6. e) not monotonic; bounded below by 1 and above by 11/12|.
The sequence a_n = 6n + (-1)^n/6n is non-monotonic and bounded below by 5/6 and above by 13/12. So, the correct answer is A).
We observe that the sequence can be written as[tex]$a_n = \frac{6n}{|6n|} + \frac{(-1)^n}{6n} = \frac{6n}{|6n|} + \frac{(-1)^n}{6|n|}.$[/tex]
We have [tex]$a_{2n} = \frac{12n}{6n} + \frac{1}{6n} = \frac{13}{6} \leq \frac{13}{6}$[/tex] and [tex]$a_{2n+1} = \frac{-12n-6}{6n+3} - \frac{1}{6n+3} = -\frac{13}{12} \geq -\frac{13}{12}.$[/tex]Therefore, the sequence is increasing and bounded below by 5/6 and above by 13/12.
We have[tex]$a_{2n} = \frac{12n}{6n} + \frac{1}{6n} = \frac{13}{6} \geq \frac{0}{1}$[/tex]and
[tex]$a_{2n+1} = \frac{-12n-6}{6n+3} - \frac{1}{6n+3} = -\frac{13}{12} \geq -\frac{13}{12}.$[/tex] Therefore, the sequence is non-increasing and bounded below by 0 and above by 6.
From above part, we see that the sequence is not monotonic.
We have [tex]$a_{2n} = \frac{12n}{6n} + \frac{1}{6n} = \frac{13}{6} \geq 1$[/tex] and[tex]$a_{2n+1} = \frac{-12n-6}{6n+3} - \frac{1}{6n+3} = -\frac{13}{12} \leq \frac{13}{12}.$[/tex] Therefore, the sequence is decreasing and bounded below by 1 and above by 6.
We have [tex]$a_{2n} = \frac{12n}{6n} + \frac{1}{6n} = \frac{13}{6} \geq 1$[/tex] and [tex]$a_{2n+1} = \frac{-12n-6}{6n+3} - \frac{1}{6n+3} = -\frac{13}{12} \geq \frac{-11}{12}.$[/tex]Therefore, the sequence is not monotonic and bounded below by 1 and above by 11/12.
Therefore, the answer is a_n = 6n + (-1)^n/6n| is increasing; bounded below by 5/6 and above by 13/12. So, the correct option is A).
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let a be a 5x4 matrix. what must a and b be if we define the linear transformation by t:ra -> rb sd t(x)=ax
If we define the linear transformation t: Ra -> Rb by t(x) = ax, where a is a 5x4 matrix, then the dimensions of the vectors in Ra and Rb will depend on the number of columns of the matrix a.
In order for the transformation to be defined, the number of columns in a must be equal to the dimension of the vectors in Ra. Therefore, if Ra is a vector space of dimension 4, then a must be a 5x4 matrix.
To determine the dimensions of Rb, we need to consider the effect of the transformation on the vectors in Ra. Since t(x) = ax, the output of the transformation will be a linear combination of the columns of a, with coefficients given by the entries of x. Therefore, the dimension of Rb will be equal to the number of linearly independent columns of a.
In order to determine b, we need to know the dimension of Rb. Once we know the dimension, we can choose any basis for Rb and represent any vector in Rb as a linear combination of the basis vectors. Then, we can solve for the coefficients of the linear combination using the inverse of a, if it exists. Therefore, the choice of b depends on the choice of basis for Rb.
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17.
An object is shot upward and it moves in a parabola path. The path is given by the
quadratic function f(x) = 30x - 5x².
(a) Express it in the form of a(x - p)² + q where a, p and q are constant.
(b) Find the maximum height of the object.
(a)The function in vertex form is f(x) = -5(x-3)² + 45, where a=-5, p=3, and q=45.
(b) The maximum height of the object occurs at the vertex of the parabola. In this case, the vertex is at (3, 45). Thus, the maximum height of the object is 45 units.
What is parabola?A parabola is a symmetrical, U-shaped curve that is formed by the graph of a quadratic function. It is a type of conic section, which can be formed by intersecting a cone with a plane that is parallel to one of its sides. The parabola has many important applications in mathematics and physics, including projectile motion, optics, and the study of gravitational fields.
(b) The maximum height of the object occurs at the vertex of the parabola. In this case, the vertex is at (3, 45). Thus, the maximum height of the object is 45 units.
What is quadratic function?A quadratic function is a function that can be written in the form f(x) = ax²+ bx + c, where a, b, and c are constants and a is not equal to zero. The graph of a quadratic function is a parabola, which is a symmetrical U-shaped curve.
To express the function in vertex form, we need to complete the square:
f(x) = -5x² + 30x
f(x) = -5(x² - 6x)
f(x) = -5(x² - 6x + 9 - 9)
f(x) = -5((x-3)² - 9)
f(x) = -5(x-3)² + 45
The function in vertex form is f(x) = -5(x-3)² + 45, where a=-5, p=3, and q=45.
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I will GIVE BRAINLIEST
Answer:
a and c are correct.
Step-by-step explanation:
In this arithmetic sequence, the first term is 3, and the common difference is 2. So a and c are correct.
a(n) = 3 + 2n for n>0 since a(0) = 3
c(n) = -1 + 2n for n>2 since c(2) = 3
verify that the intermediate value theorem applies to the indicated interval and find the value of c guaranteed by the theorem. f(x) = x2 3x 2, [0, 5], f(c) = 20
Answer:
Step-by-step explanation:
To apply the intermediate value theorem, we need to show that the function f(x) = x^2 + 3x + 2 is continuous on the closed interval [0, 5].
Since f(x) is a polynomial function, it is continuous on the entire real line. Therefore, it is also continuous on the closed interval [0, 5].
To find the value of c guaranteed by the theorem, we need to find two values a and b in [0, 5] such that f(a) < 20 < f(b).
We have:
f(0) = 2
f(5) = 60
Since f(x) is an increasing function on [0, 5], we can conclude that for any value of x between 0 and 5, f(x) will lie between f(0) and f(5).
Therefore, there exists a value c in [0, 5] such that f(c) = 20.
We have verified that the intermediate value theorem applies to the given function on the interval [0, 5] and the value of c guaranteed by the theorem is a solution of f(c) = 20.
After taking part in a competition, Adriana received a bronze medal with a diameter of 6 centimeters. What is the medal's radius?
Answer:
3
Step-by-step explanation:the diameter is twice as long as the radius, therefore you need to half the diameter for the radius
Answer:
3
Step-by-step explanation:
[tex]r=\frac{d}{2}[/tex], where r is the radius and d is the diameter. Since the diameter is 6, [tex]\frac{6}{2} =3[/tex], which means the radius is 3.
a dosage strenfght pf 0.2 mg in 1.5ml is give 0.15mg
A "dosage-strength" of "0.2-mg" in "1.5-mL" is available. Give 0.15 mg. in 1.125 mL.
The "Dosage-Strength" is defined as the concentration of a medication, generally expressed in terms of the amount of active ingredient(s) present per unit of volume or weight.
To calculate the volume of the 0.2 mg dosage strength needed to obtain 0.15 mg, we use the following formula:
⇒ Volume to withdraw = (Dosage needed/Dosage strength) × Volume of available dosage strength,
Substituting the values,
We get,
⇒ Dosage needed = 0.15 mg,
⇒ Dosage strength = 0.2 mg,
⇒ Volume of 0.2 mg = 1.5 mL,
So, Volume of 0.15 mg = (0.15 mg/0.2 mg) × 1.5 mL,
⇒ 1.125 mL.
Therefore, 0.15 mg of the medication can be obtained by using 1.125 mL of the available 0.2 mg dosage strength.
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The given question is incomplete, the complete question is
A dosage strength of 0.2 mg in 1.5 mL is available. Give 0.15 mg. in ___ mL.
Find the prime factorization of each of these integers and use each factorization to answer the questions posed. The greatest prime factor of 39 is _____.
The prime factorization of 39 is 3 × 13. Therefore, the greatest prime factor of 39 is 13.
The prime factorization of a number involves breaking it down into its prime factors, which are the prime numbers that multiply together to give the original number. Here's how the prime factorization of 39 is calculated:
Start with the number 39.
Find the smallest prime number that divides evenly into 39. In this case, it's 3, because 3 x 13 = 39.
Divide 39 by 3 to get the quotient of 13.
Since 13 is a prime number, it cannot be divided any further.
Write the prime factors in ascending order: 3 x 13.
So, the prime factorization of 39 is 3 x 13. This means that 39 can be expressed as the product of 3 and 13, both of which are prime numbers.
Now, to determine the greatest prime factor of 39, we simply look at the prime factors we obtained, which are 3 and 13. Since 13 is larger than 3, it is the greatest prime factor of 39. Therefore, the statement "the greatest prime factor of 39 is 13" is correct based on the prime factorization of 39 as 3 x 13.
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Write the first five terms of the recursively defined sequence. a1= 10, ak +1-5 ak a1 =110 a2 = 20 a3 = 40 a4 = itq : : 1()
The first five terms of the recursively defined sequence a1= 10, ak +1-5 ak a1 =110 a2 = 20 a3 = 40 a4 = itq are a1 = 10 ,a2 = 20,a3 = 40,a4 = 180 and a5 = 440 .
To find each term in the series, we use the recursive formula:
ak+1 = 5ak - a1
Starting with a1 = 10, we can find a2:
a2 = 5a1 - a1 = 4a1 = 40
Using a2, we can find a3:
a3 = 5a2 - a1 = 5(40) - 10 = 190
Using a3, we can find a4:
a4 = 5a3 - a1 = 5(190) - 10 = 940
And using a4, we can find a5:
a5 = 5a4 - a1 = 5(940) - 10 = 4690
Therefore, the first five terms of the sequence are 10, 20, 40, 180, and 440.
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Simplify. y^2/y^7 please hurry I need help with this stuff
Answer:
1/y^5.
Step-by-step explanation:
To simplify y²/y⁷, we can use the quotient rule of exponents, which states that when dividing exponential terms with the same base, we can subtract the exponents. Specifically, we have:
y²/y⁷ = y^(2-7) = y^(-5)
Now, we can simplify further by using the negative exponent rule, which states that a term with a negative exponent is equal to the reciprocal of the same term with a positive exponent. Specifically, we have:
y^(-5) = 1/y^5
Therefore, y²/y⁷ simplifies to 1/y^5.
usign he sepreaion of variavbles echinuqe solve the following differetiablw equation with initial conditions: dy/dx=e^(2x 3y) and y(0)=1 (Hint: Use a property of exponentials to rewrite the differential equation so it can be separated.) The solution is:
The solution to the differential equation dy/dx = e^(2x 3y) with initial condition y(0) = 1 is: y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³
To solve the differential equation dy/dx = e^(2x 3y) using separation of variables, we first need to rewrite it in a separable form. Using the property of exponentials that e^(a+b) = eᵃ × eᵇ, we can rewrite the equation as:
1/y dy = e^(2x) dx × e^(3y)
Now we can separate the variables by integrating both sides:
∫(1/y) dy = ∫(e^(2x) dx × e^(3y))
ln|y| = (1/2)e^(2x) × e^(3y) + C
where C is the constant of integration.
Applying the initial condition y(0) = 1, we can solve for C:
ln|1| = (1/2)e^(2×0) × e^(3*1) + C
0 = (1/2) × e³ + C
C = -1/2 × e³
Substituting C back into the equation, we get:
ln|y| = (1/2)e^(2x) × e^(3y) - 1/2 × e³
Simplifying and solving for y, we get:
y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³
Therefore, the solution to the differential equation dy/dx = e^(2x 3y) with initial condition y(0) = 1 is:
y = (1/3) ln|e⁶ˣ - 1| - (1/6)e³
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Evaluate the expression 7 + 2 x 8 − 5. (1 point)
18
20
48
Answer:
The correct answer would be 18
Step-by-step explanation:
assume the random variable x is normally distributed with mean 83 and standard deviation 4 . find the indicated probability. p(x<77))
The probability of x being less than 77 is approximately 0.0668 or 6.68%.
To solve this problem, we need to standardize the variable x to the standard normal distribution with a mean of 0 and a standard deviation of 1. We can do this using the formula:
z = (x - mu) / sigma
where z is the standard score, x is the variable of interest, mu is the mean, and sigma is the standard deviation.
Substituting the given values, we get:
z = (77 - 83) / 4 = -1.5
Now we need to find the probability that a standard normal variable is less than -1.5. We can use a standard normal table or a calculator to find that:
P(z < -1.5) = 0.0668
Therefore, the probability that x is less than 77 is:
P(x < 77) = P(z < -1.5) = 0.0668
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Suppose that the random variable X is the time taken by a garage to service a car. These times are distributed between 0 and 10 hours with a cumulative distribution function
F (x) = A + B ln(3x + 2) for 0 ≤ x ≤ 10.
(a) Find the values of A and B and sketch the cumulative distribution function.
(b) What is the probability that a repair job takes longer than two hours?
(c) Construct and sketch the probability density function.
(a) The values of: A = ln(32) / (ln(32) - ln(6)); B = -1 / (ln(32) - ln(6)) and the cumulative distribution function is F(x) = ln(32) / (ln(32) - ln(6)) - [1 / (ln(32) - ln(6))] ln(3x + 2). (b) The probability is 0.102. (c) The probability density function is only defined on the interval [0, 10].
(a) Since F(x) is a cumulative distribution function, we have:
lim x→0 F(x) = 0
lim x→10 F(x) = 1
Using these limits:
lim x→0 F(x) = A + B ln(3x + 2) = 0
A = -B ln(6)
lim x→10 F(x) = A + B ln(3x + 2) = 1
A + B ln(32) = 1
-B ln(6) + B ln(32) = 1 - A
B = -1 / (ln(32) - ln(6))
A = -B ln(6) = ln(32) / (ln(32) - ln(6))
The cumulative distribution function is:
F(x) = ln(32) / (ln(32) - ln(6)) - [1 / (ln(32) - ln(6))] ln(3x + 2)
(b) The probability that a repair job takes longer than two hours is:
P(X > 2) = 1 - P(X ≤ 2) = 1 - F(2) = 1 - ln(32) / (ln(32) - ln(6)) + [1 / (ln(32) - ln(6))] ln(8)
≈ 0.102
(c) To find the probability density function f(x), we differentiate F(x):
f(x) = d/dx F(x) = [3 / ((3x + 2) ln(2))] / (ln(32) - ln(6))
The function is only defined on the interval [0, 10]. The graph of f(x) is decreasing on [0, 2] and increasing on [2, 10].
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Put the numbers in each category to which they belong.
1) a rational number is -2/5
2) -14/9 is a rational number
3) 567 is a prime number, a whole number
4) -20/5 is a rational number, an integer
What is a rational number, a whole number, and a prime number?Rational numbers are any numbers that can be expressed as p/q, where p and q are integers and q is not equal to zero. Whole Numbers- Whole numbers are integers ranging from 0 to infinity. Prime numbers are those that have only 1 and themselves as factors.
A rational number is -2/5. It is a fraction with a numerator of -2 and a denominator of 5.
-14/9 is a rational number. It is a fraction with a numerator of -14 and a denominator of 9.
567 is a prime number. It is a whole number as well as an integer. It cannot be stated as a fraction with a denominator other than one, hence it is not a rational number.
-20/5 is a sensible number. It is the same as -4, which is an integer. It is also an even number. Because it is negative, it is not a natural number.
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In a small lottery, 10 tickets-numbered 1, 2,..., 10- are sold. Two numbers are drawn at random for prizes. You hold tickets numbered 1 and 2. What is the probability that you win at least one prize? hint: use complementation rule
The probability that you win at least one prize is 17/45.
How to find the probability that you win at least one prize?The total number of ways to draw two numbers from 10 is given by the combination formula:
C(10,2) = 10!/((10-2)!*2!) = 45
This means there are 45 possible outcomes for the lottery drawing.
The number of ways to draw two numbers from the remaining 8 tickets (excluding tickets numbered 1 and 2) is given by:
C(8,2) = 8!/((8-2)!*2!) = 28
This means that there are 28 outcomes in which neither of your tickets win a prize.
So the probability that you win at least one prize is equal to 1 minus the probability that you win no prizes:
P(win at least one prize) = 1 - P(win no prize) = 1 - 28/45 = 17/45
Therefore, the probability that you win at least one prize is 17/45.
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Please help!!
I used law of sines and put it in calculator but the answer was weird...
any help would be appreciated as this is due tomorrow!
Thank you!
m/4 =
m/5=
m/1 =
m/3 =
m/2=
m/6=
m/7=
Here are the angles and their values:
m∠1 = 63.5°m∠2 = 124°m∠3 = 29.5°m∠4 = 90°m∠5 = 54°m∠6 = 116.5°m∠7 = 121°m∠8 = 90°How to solveThese angles were found using the following properties and calculations:
The sum of the internal angles of a triangle is 180°.
The angle rotated from point B to point E (angle 7) is the sum of the angles of arcs BA and AE.
In isosceles triangles, the angles opposite equal sides are equal.
A straight line has an angle of 180°.
The angle formed by a tangent line and a radius at the point of contact is 90°.
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PLEASE HELP 30 POINTS SOLVE FOR MISSING SIDE SHOW WORK PLS
Answer:
Step-by-step explanation:
The third side of the triangle may be found using Pythagoras theorem.
Pythagoras theorem is for right angle triangles-
c^2 = a^2 + b^2
‘c’ = the side opposite the triangle’s 90 degree angle. This is called the hypotenuse.
‘a’ and ‘b’ = a and b are just the two remaining sides of the triangle that is NOT the hypotenuse. It does not matter which side you pick out of the two for ‘a’ and which side you pick for ‘b’
The side we are trying to find is the hypotenuse, which is ‘c’
Let’s say that
a= 9m
b= 12m
Substituting that into the
c^2 = a^2 + b^2 formula,
c^2 = 9^2 + 12^2
= 81+ 144
= 225
( square root both sides of the equation so we get just the value of ‘c’)
c= 15m
Therefore the missing side (the right angle triangle’s hypotenuse) is 15m.
:)
Anita has$ 800 in her savings account that earns 12% annually. The interest is not compounded. How much interest will she earn in 2 year?
Answer:$192
Step-by-step explanation:
Step 1: Multiply 800 times 12%. You get $96
Step 2: Since the question asked how much she will earn in 2yrs double the interest amount
Step 3 96+96=192