Answer:
given y=2.69+0.0138x the slope is the coefficient of x slope=0.0138 this means that the dive duration is expected to increase by about 0.0138 per minute
Step-by-step explanation:
Answer:
Hi!
A) The intercept of the regression line is 2.56.
B)The slope of the regression line is 0.0135.
C)For every increase of 1-meter depth, the mean dive duration increases by exactly 0.0135 minutes.
Explanation:
The definition of the regression line of a function is the value of Y(x) when x=0.
The slope of a function is the value that configures the ratio of change vertically on the regression.
In this case is represented by 0.0135 multiplied by the x.
If we are talking about depth, and the values of x are all positives values.
Y(50)=2.56+0.0135(50)=3.235
So, when the penguins go deeper, they remain more time diving.
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t2 + 35 , y = ln(t2 + 35), z = t; (6, ln(36), 1)
Step-by-step explanation:
i think the answer is 42 to be exacr
A process produces a certain part with a mean diameter of 2 inches and a standard deviation of 0.05 inches. The lower and upper engineering specification limits are 1.6 inches and 3.5 inches. What is the Cp (measure of potential capability)?
Answer:
The value of Cp (measure of potential capability) is 6.33.
Step-by-step explanation:
Given information: Process average = 2 inches, process standard deviation = 0.05 inches, lower engineering specification limit = 1.6 inches and upper engineering specification limit =3.5 inches.
The formula for Cp (measure of potential capability) is
[tex]CP=\frac{USL-LSL}{6\sigma}[/tex]
Where, USL is upper specification limit, LSL is specification limit, σ is process standard deviation.
Substitute USL=3.5, LSL=1.6 and σ=0.05 in the above formula.
[tex]CP=\frac{3.5-1.6}{6(0.05)}[/tex]
[tex]CP=\frac{1.9}{0.3}[/tex]
[tex]CP=6.3333[/tex]
[tex]CP\approx 6.33[/tex]
Therefore the value of Cp (measure of potential capability) is 6.33.
Find the lengths of the sides of the triangle PQR. P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1) |PQ| = Correct: Your answer is correct. |QR| = Correct: Your answer is correct. |RP| = Correct: Your answer is correct. Is it a right triangle? Yes No Is it an isosceles triangle? Yes No
Answer:
1. The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.
2. No.
3. No.
Step-by-step explanation:
The vertices of given triangle are P(0, 1, 5), Q(2, 3, 4), R(2, −3, 1).
Distance formula:
[tex]D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}[/tex]
Using distance formula we get
[tex]|PQ|=\sqrt{(2-0)^2+(3-1)^2+(4-5)^2}=\sqrt{9}=3[/tex]
[tex]|QR|=\sqrt{(2-2)^2+(-3-3)^2+(1-4)^2}=\sqrt{45}=3\sqrt{5}[/tex]
[tex]|RP|=\sqrt{(0-2)^2+(1-(-3))^2+(5-1)^2}=\sqrt{36}=6[/tex]
The values of |PQ|, |QR| and |RP| are 3, 3√5 and 6 respectively.
In a right angled triangle the sum of squares of two small sides is equal to the square of third side.
[tex](3)^2+(3\sqrt{5})^2=54\neq 6^2[/tex]
Therefore PQR is not a right angled triangle.
In an isosceles triangle, the length of two sides are equal.
The measure of all sides are different, therefore PQR is not an isosceles triangle.
Find an equation of the line containing the given pair of points. (3,5) and (9,6) y- (Simplify your answer. Type your answer in slope-intercept form. Use integers or fract
Answer: [tex]y=\dfrac{1}{3}x+4[/tex]
Step-by-step explanation:
The equation of a line passing through (a,b) and (c,d) is given by :_
[tex](y-b)=\dfrac{d-b}{c-a}(x-a)[/tex]
The given points : (3,5) and (9,6)
Then , the equation of a line passing through (3,5) and (9,6) will be :-
[tex]y-5=\dfrac{6-5}{9-3}(x-3)\\\\\Rightarrow\ y-5=\dfrac{1}{3}(x-3)\\\\\ y=\dfrac{1}{3}x-1+5\\\\\Rightarrow\ y=\dfrac{1}{3}x+4[/tex]
Hence, the equation of the line in slope -intercept form : [tex]y=\dfrac{1}{3}x+4[/tex]
Final answer:
The answer provides the equation of the line passing through the points (3,5) and (9,6) in slope-intercept form.Using the points (3,5) and (9,6), the change in [tex]\( y \) is \( 6 - 5 = 1 \) and the change in \( x \) is \( 9 - 3 = 6 \). So, the slope is \( \frac{1}{6} \).[/tex]
Explanation:
Equation of the line:
The slope of a line represents the rate of change between two points on the line. It indicates how much the dependent variable (y-coordinate) changes for a given change in the independent variable (x-coordinate).
In this case, given the two points (3,5) and (9,6), we can calculate the slope using the formula:
[tex]\[ \text{slope} = \frac{{\text{change in } y}}{{\text{change in } x}} \][/tex]
Using the points (3,5) and (9,6), the change in [tex]\( y \) is \( 6 - 5 = 1 \) and the change in \( x \) is \( 9 - 3 = 6 \). So, the slope is \( \frac{1}{6} \).[/tex]
Solve the separable initial value problem. 1. y' = ln(x)(1 + y2), y(1) = 3 = y= tan(xlnx-x+1+arctan(3) 2. y' = 9x? V1 + x? (1 + y2), y(0) = 3 = y=
The solution to the separable initial value problem is [tex]\( y = \tan(x \ln(x) - x + 1 + \arctan(3)) \).[/tex]
Explanation:A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a function's rate of change is related to its current value and possibly other variables. Differential equations are used to model various physical, biological, and social phenomena in fields such as physics, engineering, biology, and economics.
To solve the differential equation [tex]\( y' = \ln(x)(1 + y^2) \)[/tex] with the initial condition [tex]\( y(1) = 3 \)[/tex], we separate variables and integrate both sides.
After integration, we get [tex]\( \tan(y) = x\ln(x) - x + C \)[/tex], where [tex]\( C \)[/tex] is the constant of integration. Using the initial condition, we find [tex]\( C = 1 + \arctan(3) \)[/tex].
Substituting this value back into the equation, we obtain the solution[tex]\( y = \tan(x \ln(x) - x + 1 + \arctan(3)) \).[/tex]This function satisfies the given differential equation and initial condition.
Sara buys a house for $290,000. She makes a 20% down payment and finances the balance with a 30-year fixed loan at 4.2% interest compounded monthly. Sara's monthly payment is:
Answer:
monthly payment=$322.52
Step-by-step explanation:
cost of house=$290,000
down payment= 20%
interest monthly = 4.2%
interest rate compounded monthly so (i)=4.2/12=0.35%
months= [tex]30\times 12[/tex]=360 months
down payment = [tex]0.2\times 290000[/tex]
=$58000
amount to be paid(P)=$232,000
[tex]P=R\frac{(1+r)^n-1}{i}\\232000=R\frac{(1+0.0035)^{360}-1}{0.0035}\\232000=R\times 719.33[/tex]
R=$322.52
sara's monthly payment will be $322.52
amir gupta’s car showroom is giving special offer on one model. their advertised price for this model is four consecutive quarters was $10450 , $10800, $11450, and $9999. use the fourth quarters as a base period. calculate the price index and percentage point rise between each quarters
Answer:
Step-by-step explanation:
S.NO QUATERS PRICE ($) PRICE RELATIVES
[tex]I = \frac{q_i}{q_4} *100[/tex]
1 q_1 10450 104.51
2 q_2 10800 108.01
3 q_3 11450 114.51
4 q_2 9999 100.00
Price Index is given as [tex]= \frac{\sum I}{n}[/tex]
[tex] = \frac{104.51+108.01+114.51+100}{4}[/tex]
= 106.75
b) percentage point rise
[tex]for q_1 = \frac{q_2 -q_1}{q_1}*100[/tex]
[tex]= \frac{108.01-104.51}{104.51}[/tex]
= 3.34%
[tex]for q_2 = \frac{q_3 -q_2}{q_2}[/tex]
[tex]= \frac{114.51-108.01}{108.01} *100[/tex]
= 6.01%
[tex]for q_3 = \frac{q_4 -q_3}{q_3}[/tex]
[tex]= \frac{100-114.51}{114.51} *100[/tex]
= 12.67%
The Royal Fruit Company produces two types of fruit drinks. The first type is 60% pure fruit juice, and the second type is 85% pure fruit juice. The company is attempting to produce a fruit drink that contains 80% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 160 pints of a mixture that is 80% pure fruit juice?
Answer: There are 32 pints of first type and 128 pints of second type in mixture.
Step-by-step explanation:
Since we have given that
Percentage of pure fruit juice in first type = 60%
Percentage of pure fruit juice in second type = 85%
Percentage of pure fruit juice in mixture = 80%
We will use "Mixture and Allegation" to find the ratio of first and second type in mixture:
First type Second type
60% 85%
80%
------------------------------------------------------------------------
85-80 : 80-60
5% : 20%
1 : 4
so, the ratio of first and second type is 1:4.
Total number of pints of mixture = 160
Number of pints of mixture of first type in mixture is given by
[tex]\dfrac{1}{5}\times 160\\\\=32\ pints[/tex]
Number of pints of mixture of second type in mixture is given by
[tex]\dfrac{4}{5}\times 160\\\\=4\times 32\\\\=128\ pints[/tex]
Hence, there are 32 pints of first type and 128 pints of second type in mixture.
Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter.x = 6 + ln(t), y = t2 + 6, (6, 7)
The slope of the tangent line to the curve defined by x = 6 + ln(t), y = t^2 + 6 at the point (6,7) can be found by differentiating x and y with respect to t and then substituting t = 1. The equation of the tangent line is y = 2x -5.
Explanation:To find the equation of the tangent to the curve at the given point, we will first need to find the derivative (slope) at the given point. The equations given are x = 6 + ln(t) and y = t2 + 6. Given point is (6, 7).
Without eliminating the parameter, we differentiate both x and y with respect to t. This allows us to find dx/dt = 1/t and dy/dt = 2t. The slope of the tangent line at (6, 7) is then (dy/dt) / (dx/dt) = 2t * t = 2*t2.
Substitute the given point (6,7) into x = 6 + ln(t), to obtain t = e0 = 1. Therefore, the slope of the tangent line is 2*12 = 2.
The equation of the tangent line can be written as: (y - y1) = m*(x - x1), where m = 2 is the slope, and (x1, y1) is the given point (6, 7).Substitute these into the equation, gets us: y-7 = 2*(x - 6), which can be simplified to: y = 2x -5.
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The equation of the tangent to the curve at the point (6, 7) is y = 2x - 5.
Explanation:To find the equation of the tangent to the curve without eliminating the parameter, we can use the parametric equations: x = 6 + ln(t) and y = t^2 + 6.
First, we need to find the derivative of y with respect to x and evaluate it at the given point (6, 7).
The derivative of y with respect to x is dy/dx = (dy/dt)/(dx/dt).
From the given equations, we can calculate dx/dt = 1/t and dy/dt = 2t.
Substituting these values into the derivative expression, we have dy/dx = (2t)/(1/t) = 2t^2.
Now, substitute the given x-coordinate (6) into the equation for x to find the corresponding t-value: 6 = 6 + ln(t) => ln(t) = 0 => t = 1.
Now, substitute the t-value (1) into the equation for y to find the corresponding y-coordinate: y = 1^2 + 6 = 7.
Therefore, the slope of the tangent at the point (6, 7) is 2(1)^2 = 2.
Using the point-slope form of a line, we can write the equation of the tangent line: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the tangent.
Plugging in the values, we have y - 7 = 2(x - 6).
Simplifying the equation, we get y = 2x - 5.
Therefore, the equation of the tangent to the curve at the point (6, 7) is y = 2x - 5.
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-56 + _ =-84
please help
Answer:
-28 is the answer.
Step-by-step explanation:
84-56=28
56+28=84
-56+-28=-84
Answer:
[tex] - 56 + x = - 84 \\ x = 56 - 84 \\ \boxed{ x = - 28}[/tex]
x= -28 is the right answer.Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After 9 hours of burning, a candle has a height of 17.5 centimeters. After 24 hours of burning, its height is 22 centimeters. What is the height of the candle after 22 hours?
Answer:
The candle has a height of 21.4 cm after burning for 22 hours.
Step-by-step explanation:
let x=hours, m=rate of change, and y= candle height
First you have to find the slope or, rate of change using the slope formula. y2-y1 divided by x2-x1 .
Here is our points (9, 17.5) and (24, 22)
x1 y1 x2 y2
Now we put these into the equation and solve
[tex]\frac{22-17.5}{24-9}[/tex] =[tex]\frac{3}{10}[/tex]
Now that we have the slope of 3/10 we can use this to find the y-intercept using the point-slope equation.
[tex]y-y_{1} =m(x-x_{1} )[/tex] y-17.5= .3(x-9) Solve
y-17.5=.3x-2.7 y -14.8= .3x
+2.7 +2.7 +14.8 +14.8
y=.3x+14.8 the y-intercept is 14.8
Now we use this equation to plug in the 22 hours.
y=.3(22) +14.8
y=6.6+14.8
y= 21.4 The candle has a height of 21.4 cm after burning for 22 hours.
A company produces a women's bowling ball that is supposed to weigh exactly 14 pounds. Unfortunately, the company has a problem with the variability of the weight. In a sample of 7 of the bowling balls the sample standard deviation was found to be 0.64 pounds. Construct a 95% confidence interval for the variance of the bowling ball weight. Assume normality. a) What is the lower limit of the 95% interval? Give your answer to three decimal places. b) What is the upper limit of the 95% interval? Give your answer to three decimal places. c) Which of the following assumptions should be checked before constructing the above confidence interval? the data need to follow a normal distribution the data need to have small variance the data need to follow a chi-square distribution the data need to be right skewed
Answer:
Step-by-step explanation:
Mean = 14
Std deviation of sample s = 0.64
n = sample size =7
Std error = [tex]\frac{s}{\sqrt{n} } =0.2419[/tex]
t critical for 95% two tailed = 2.02
Margin of error = 2.02*SE = 0.4886
a)Conf interval lower bound = 14-0.4886 = 13.5114
b)Upper bound = 14+0.4886 = 14.4886
c)Assumption
the data need to follow a normal distribution
Researchers sampled 178 young women who recently participated in a STEM program. Of the 178 STEM participants, 110 were in urban areas, 56 in suburban areas, and 12 in rural areas. If one of the participants is selected at random, what is the probability that she is from an urban area? Not a rural area?
Answer:
Probability from Urban Area = [tex]\frac{55}{89}[/tex]
Probability NOT from Rural Area = [tex]\frac{83}{89}[/tex]
Step-by-step explanation:
Total 178
Urban 110
Suburban 56
Rural 12
Hence, probability of x is number of x divided by total.
So, probability that she is from an urban area = 110/178 = 55/89
And
probability NOT a rural area (means urban and suburban which is 110+56=166) = 166/178 = 83/89
To calculate the probability of selecting a participant from an urban area, divide the number of urban participants (110) by the total number of participants (178), yielding approximately 0.61798. For the probability of not a rural area, sum the urban and suburban participants (110 + 56) and divide by the total, which gives approximately 0.93258.
The question asks about finding the probability of a participant being from an urban area and not from a rural area in a sample of young women in a STEM program.
The total number of participants is 178. Of these, 110 are from urban areas. To find the probability of selecting a participant from an urban area, we divide the number of urban participants by the total number of participants:
Probability (Urban) = Number of Urban Participants / Total Number of Participants = 110 / 178 ≈ 0.61798
Similarly, to find the probability of not selecting a participant from a rural area, we need to first find the number of participants who are not from rural areas. This is the sum of urban and suburban participants, or 110 + 56. Then we calculate:
Probability (Not Rural) = Number of Non-Rural Participants / Total Number of Participants = (110 + 56) / 178 ≈ 0.93258
Use a Taylor Series solution, centered at zero to solve the initial-value problem below. (Find a 5th degree approximation only) dy/dx = x + y y(0) = 1
We're looking for a solution of the form
[tex]y=\displaystyle\sum_{n=0}^\infty a_nx^n=a_0+a_1x+a_2x^2+\cdots[/tex]
Given that [tex]y(0)=1[/tex], we would end up with [tex]a_0=1[/tex].
Its first derivative is
[tex]y'=\displaystyle\sum_{n=0}^\infty na_nx^{n-1}=\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n[/tex]
The shifting of the index here is useful in the next step. Substituting these series into the ODE gives
[tex]\displaystyle\sum_{n=0}^\infty(n+1)a_{n+1}x^n-\sum_{n=0}^\infty a_nx^n=x[/tex]
Both series start with the same-degree term [tex]x^0[/tex], so we can condense the left side into one series.
[tex]\displaystyle\sum_{n=0}^\infty\bigg((n+1)a_{n+1}-a_n\bigg)x^n=x[/tex]
Pull out the first two terms ([tex]x^0[/tex] and [tex]x^1[/tex]) of the series:
[tex]a_1-a_0+(2a_2-a_1)x+\displaystyle\sum_{n=2}^\infty\bigg((n+1)a_{n+1}-a_n\bigg)x^n=x[/tex]
Matching the coefficients of the [tex]x^0[/tex] and [tex]x^1[/tex] terms on either side tells us that
[tex]\begin{cases}a_1-a_0=0\\2a_2-a_1=1\end{cases}[/tex]
We know that [tex]a_0=1[/tex], so [tex]a_1=1[/tex] and [tex]a_2=1[/tex]. The rest of the coefficients, for [tex]n\ge2[/tex], are given according to the recurrence,
[tex](n+1)a_{n+1}-a_n=0\implies a_{n+1}=\dfrac{a_n}{n+1}[/tex]
so that [tex]a_3=\dfrac{a_2}3=\dfrac13[/tex], [tex]a_4=\dfrac{a_3}4=\dfrac1{12}[/tex], and [tex]a_5=\dfrac{a_4}5=\dfrac1{60}[/tex]. So the 5th degree approximation to the solution to this ODE centered at [tex]x=0[/tex] is
[tex]y\approx1+x+x^2+\dfrac{x^3}3+\dfrac{x^4}{12}+\dfrac{x^5}{60}[/tex]
70% of the workers at Motor Works are female, while 68% of the workers at City Bank are female. If one of these companies is selected at random (assume a 50-50 chance for each), and then a worker is selected at random, what is the probability that the worker will be female?
Answer:
69%
Step-by-step explanation:
If Motor Works is selected, the probability that the worker is a female is 70%. If City Bank is selected, the probability is 68%.
But we don't know what company will be selected, we only know that they have the same probability, 50-50.
So, with 50% of probability the worker will be female with a 70% of probability (because they selected from Motor Works) and with 50% of probability the worker will be female with a 68% of probability (they selected from City Bank).
We express this as 50%*70% + 50%*68% = 69%
The multiplication means that both probabilities happen together and the sum means that happens one thing or the other (they select Motor Works or City Bank)
Final answer:
The probability that a randomly selected worker from either Motor Works or City Bank is female is 0.69, or 69%.
Explanation:
To calculate the probability that a randomly selected worker is female, we need to consider the probability of selecting each company and then the probability of selecting a female worker from that company. We are given that the probability of selecting Motor Works or City Bank is equal, hence it is 1/2 for each. Now, let's calculate the overall probability using the following steps:
Calculate the probability of selecting a female from Motor Works: P(Female at Motor Works) = Probability of Motor Works times Probability of female at Motor Works = 1/2 times 70% = 0.35.
Calculate the probability of selecting a female from City Bank: P(Female at City Bank) = Probability of City Bank times Probability of female at City Bank = 1/2 times 68% = 0.34.
The total probability of selecting a female from either company is the sum of these individual probabilities: P(Female) = P(Female at Motor Works) + P(Female at City Bank) = 0.35 + 0.34 = 0.69.
The probability that a randomly selected worker from either company is female is therefore 0.69, or 69%.
Suppose a > 0 is constant and consider the parameteric surface sigma given by r(phi, theta) = a sin(phi) cos(theta)i + a sin(phi) j + a cos(phi) k. 0 lessthanorequalto theta lessthanorequalto 2 pi, 0 lessthanorequalto phi lessthanorequalto pi. (a) Directly verify algebraically that r parameterizes the sphere x^2 + y^2 + z^2 = a^2, by substituting x = a sin(phi), y = a sin(phi) sin(theta), and z = a cos(phi) into the left-hand side of the equation. (b) Find r_phi, r_theta, r_phi times r_theta, and |r_phi times r_theta|. (c) Compute the surface area of the sphere doubleintegral_sigma l dS using change of variables. Find the surface area of the band sigma cut from the paraboloid z = x^2 + y^2 by the planes z = 2 and z = 6 by first finding a parameterization for the surface and then computing doubleintegral_sigma dS. Find the flux of the field F = x^2j - xzk across the surface cut by the parabolic cylinder y = x^2, -1 lessthanorequalto x lessthanorequalto 1, by the planes z = 0 and z = 2, Your normal vector should point in the direction indicated in the figure below.
[tex]\Sigma[/tex] should have parameterization
[tex]\vec r(\varphi,\theta)=a\sin\varphi\cos\theta\,\vec\imath+a\sin\varphi\sin\theta\,\vec\jmath+a\cos\varphi\,\vec k[/tex]
if it's supposed to capture the sphere of radius [tex]a[/tex] centered at the origin. ([tex]\sin\theta[/tex] is missing from the second component)
a. You should substitute [tex]x=a\sin\varphi\cos\theta[/tex] (missing [tex]\cos\theta[/tex] this time...). Then
[tex]x^2+y^2+z^2=(a\sin\varphi\cos\theta)^2+(a\sin\varphi\sin\theta)^2+(a\cos\varphi)^2[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi\cos^2\theta+\sin^2\varphi\sin^2\theta+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi\left(\cos^2\theta+\sin^2\theta\right)+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2\left(\sin^2\varphi+\cos^2\varphi\right)[/tex]
[tex]x^2+y^2+z^2=a^2[/tex]
as required.
b. We have
[tex]\vec r_\varphi=a\cos\varphi\cos\theta\,\vec\imath+a\cos\varphi\sin\theta\,\vec\jmath-a\sin\varphi\,\vec k[/tex]
[tex]\vec r_\theta=-a\sin\varphi\sin\theta\,\vec\imath+a\sin\varphi\cos\theta\,\vec\jmath[/tex]
[tex]\vec r_\varphi\times\vec r_\theta=a^2\sin^2\varphi\cos\theta\,\vec\imath+a^2\sin^2\varphi\sin\theta\,\vec\jmath+a^2\cos\varphi\sin\varphi\,\vec k[/tex]
[tex]\|\vec r_\varphi\times\vec r_\theta\|=a^2\sin\varphi[/tex]
c. The surface area of [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\mathrm dS=a^2\int_0^\pi\int_0^{2\pi}\sin\varphi\,\mathrm d\theta\,\mathrm d\varphi[/tex]
You don't need a substitution to compute this. The integration limits are constant, so you can separate the variables to get two integrals. You'd end up with
[tex]\displaystyle\iint_\Sigma\mathrm dS=4\pi a^2[/tex]
# # #
Looks like there's an altogether different question being asked now. Parameterize [tex]\Sigma[/tex] by
[tex]\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+u^2\,\vec k[/tex]
with [tex]\sqrt2\le u\le\sqrt6[/tex] and [tex]0\le v\le2\pi[/tex]. Then
[tex]\|\vec s_u\times\vec s_v\|=u\sqrt{1+4u^2}[/tex]
The surface area of [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\mathrm dS=\int_0^{2\pi}\int_{\sqrt2}^{\sqrt6}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv[/tex]
The integrand doesn't depend on [tex]v[/tex], so integration with respect to [tex]v[/tex] contributes a factor of [tex]2\pi[/tex]. Substitute [tex]w=1+4u^2[/tex] to get [tex]\mathrm dw=8u\,\mathrm du[/tex]. Then
[tex]\displaystyle\iint_\Sigma\mathrm dS=\frac\pi4\int_9^{25}\sqrt w\,\mathrm dw=\frac{49\pi}3[/tex]
# # #
Looks like yet another different question. No figure was included in your post, so I'll assume the normal vector points outward from the surface, away from the origin.
Parameterize [tex]\Sigma[/tex] by
[tex]\vec t(u,v)=u\,\vec\imath+u^2\,\vec\jmath+v\,\vec k[/tex]
with [tex]-1\le u\le1[/tex] and [tex]0\le v\le 2[/tex]. Take the normal vector to [tex]\Sigma[/tex] to be
[tex]\vec t_u\times\vec t_v=2u\,\vec\imath-\vec\jmath[/tex]
Then the flux of [tex]\vec F[/tex] across [tex]\Sigma[/tex] is
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=\int_0^2\int_{-1}^1(u^2\,\vec\jmath-uv\,\vec k)\cdot(2u\,\vec\imath-\vec\jmath)\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=-\int_0^2\int_{-1}^1u^2\,\mathrm du\,\mathrm dv[/tex]
[tex]\displaystyle\iint_\Sigma\vec F\cdot\mathrm d\vec S=-2\int_{-1}^1u^2\,\mathrm du=-\frac43[/tex]
If instead the direction is toward the origin, the flux would be positive.
Two fair, distinct dice (one red and one green) are rolled. Let A be the event the red die comes up even and B be the event the sum on the two dice is eight. Are A,B independent events?
Answer: No, A and B are not independent events.
Step-by-step explanation:
Since we have given that
Number of outcomes that a die comes up with = 6
A be the event that the red die comes up even.
A={2,4,6}
B be the event that the sum on the two dice is 8.
B={(2,4),(4,2),(5,3),(3,5),(44)}
P(A) = [tex]\dfrac{3}{6}=\dfrac{1}{2}[/tex]
P(B) = [tex]\dfrac{5}{36}[/tex]
P(A∩B) = [tex]\dfrac{3}{36}[/tex]
But,
P(A).P(B) ≠ P(A∩B)
[tex]\dfrac{1}{2}\times \dfrac{5}{36}\neq \dfrac{3}{36}\\\\\dfrac{5}{72}\neq\dfrac{1}{12}[/tex]
Hence, A and B are dependent events.
Solve the following problems:
Given: m∠DAB=m∠CBA
m∠CAB=m∠DBA
CA=13 cm
Find: DB
Answer:
DB = 13 cm
Step-by-step explanation:
ΔCAB ≅ ΔDBA by ASA, so CA ≅ DB by CPCTC.
CA = 13 cm, so DB = 13 cm.
Answer:
Step-by-step explanation:
Given : m(∠DAB) = m(∠CBA)
m (CAB) = m(∠DBA)
and CA = 13 cm
To find : measure of DB
In ΔCAB and ΔDAB
m(∠DAB) = m(∠CBA) [given]
m(∠CAB) = m(∠DBA) [given]
and AD is common in both the triangles.
Therefore, ΔCAB and ΔDAB will be congruent. [By ASA property]
Therefore, CA = DB = 13 cm.
Find X if AX+B=AC+D
Would it be (D-B/A)+C?
Answer:
it will be x=(AC-B+D)/A
Step-by-step explanation:
AX+B=AC+D
AX+B-B=AC+D-B
AX/A=(AC+D-B)/A
X=(AC+D-B)/A
x+y-2z=-9
2x-y+8z=99
x-2y+5z=23
give the solution with z arbitrary
Answer:
(x, y, z) = (-4, 29, 17)
Step-by-step explanation:
These three equations have a unique solution. If you want "z arbitrary", you need to write a system of two equations with three variables (or, equivalently, a set of dependent equations).
It is convenient to let a graphing calculator, scientific calculator, or web site solve these.
_____
You can reduce the system to two equations in y and z by ...
subtracting the last equation from the first:
3y -7z = -32
subtracting twice the last equation from the second:
3y -2z = 53
Subtracting the first of these from the second, you get ...
5z = 85
z = 17
The remaining variable values fall out:
y = (53+2z)/3 = 87/3 = 29
x = -9 +2z -y = -9 +2(17) -29 = -4
These equations have the solution (x, y, z) = (-4, 29, 17).
On July 18, Lester accepted a $15,000, 7 3/4%, 180-day note from Ryan O'Flynn. On October 5, Lester discounted the note at Brome Bank at 8 1/4%, What proceeds did Lester receive? Use ordinary interest.
Answer:
so Lester receive money is $13762.5
Step-by-step explanation:
Given data in question
principal = $15000
discount = 8* 1/4 % i.e. = 8.25% = 0.0825
to find out
Lester receive money ?
solution
we know ordinary interest formula i.e.
receive money = principal ( 1 - discount ) ...........1
we all value principal and discount in equation 1 and we get receive money
receive money = principal ( 1 - discount )
receive money = $15000 ( 1 - 0.0825)
receive money = $ 13762.5
so Lester receive money is $13762.5
Lester received a total of $14,417.50 as proceeds when he discounted the note at Brome Bank.
Explanation:Lester accepted a $15,000, 7 3/4%, 180-day note from Ryan O'Flynn on July 18. On October 5, Lester discounted the note at Brome Bank at 8 1/4%.
To calculate the proceeds Lester received, we need to find the simple interest earned on the note for 180 days. First, find the interest earned:
Principal x Rate x Time = Interest
$15,000 x 7.75% x (180/360) = $582.50
Next, subtract the interest earned from the face value of the note to find the proceeds Lester received:
$15,000 - $582.50 = $14,417.50
Therefore, Lester received $14,417.50.
Learn more about Calculating proceeds of a discounted note here:https://brainly.com/question/33296445
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The U.S. Census Bureau reports that in the year 2008, the mean household income was $68,424 and the median was $50,303. If a histogram were constructed for the incomes of all households in the United States, would you expect it to be skewed to the right, skewed to the left, or approximately symmetric?
Answer:
Skewed to the Right
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced problems.
Since the mean household income is $64,424 and the median was $50,303 then the mean is larger than the median. When this occurs then the constructed histogram is always Skewed to the Right. This is because there are a couple of really large values that affect the mean but not the middle value of the data set.
This in term leads to the graph dipping in values the farther right you go and increasing the farther left you go, as shown in the example picture below.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
find principal of a loan at 11% for 5 years with $426.25 simple interest
Answer:
[tex]P=775[/tex]
Step-by-step explanation:
The Simple Interest Equation is [tex]A = P(1 + rt)[/tex]
where
A = Total Accrued Amount (principal + interest)
P = Principal Amount
I = Interest Amount
t = Time Period involved in months or years
In this case, we do not know the values of the equation (A and P), but we know the amount of interest accrued
If we define our principal whit this formula, we are able to know the rest of the values:
[tex]A-P= interest[/tex]
clearing
[tex]A=interest+P[/tex]
replacing
[tex]426.25 + P = P (1+(0.11(5))[/tex]
Solving
[tex]426.25+P=1.55P[/tex]
[tex]P-1.55P=-426.25[/tex]
[tex]-0.55P=-426.65[/tex]
[tex]P=\frac{-426.25}{-0.55}[/tex]
[tex]P=775[/tex]
Suppose you first walk 28.6 m in a direction 20 degrees west of north and then 22 m in a direction 40 degrees south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position?
Answer:
29.5 m 64° west of north
Step-by-step explanation:
A suitable vector calculator can add the vectors for you. (See attached.) Here, we have used North as the 0° reference and positive angles in the clockwise direction (as bearings are measured).
___
You can also use a triangle solver (provided by many graphing calculators and stand-alone apps). For this, and for manual calculation (below) it is useful to realize the angle difference between the travel directions is 70°.
___
Using the Law of Cosines to find the distance from start (d), we have (in meters) ...
d² = 28.6² + 22² -2·28.6·22·cos(70°) ≈ 871.561
d ≈ √871.561 ≈ 29.52 . . . . meters
The internal angle between the initial travel direction and the direction to the end point is found using the Law of Sines:
sin(angle)/22 = sin(70°)/29.52
angle = arcsin(22/29.52×sin(70°)) ≈ 44.44°
This angle is the additional angle the destination is west of the initial travel direction, so is ...
20° west of north + 44.44° farther west of north = 64.44° west of north
__
In the second attachment, North is to the right, and West is down. This is essentially a reflection across the line y=x of the usual map directions and angles. Reflection doesn't change lengths or angles, so the computations are valid regardless of how you assign map directions to x-y coordinates.
Online jewelry sales have increased steadily. In 2003, sales were approximately 2 billion dollars, and in 2013 they were approximately 5.5 1 billion. Construct a model to predict online jewelery sales. Use your model, to find the predicted online jewelry sales for 2015? (Express your answer rounded correctly to the nearest tenth of a billion.)
Answer:
1) [tex]y=2(1+0.081)^x[/tex]
2) 6.4 billion.
Step-by-step explanation:
1) Let the model that is used to find online jewelry sales ( in billions ) after x years,
[tex]y=a(1+r)^x[/tex]
Let 2003 is the initial year,
That is, for 2003, x = 0,
The sales on 2003 is 2 billion, y = 2,
⇒ [tex]2=a(1+r)^0\implies a=2[/tex]
Now, in 2013 they were approximately 5.5 1 billion
i.e. if x = 13 then y = 5.51,
[tex]\implies 5.51 = a(1+r)^{13}[/tex]
[tex]5.51=2(1+r)^{13}[/tex]
With help of graphing calculator,
r = 0.081,
Hence, the model that represents the given scenario is,
[tex]y=2(1+0.081)^x[/tex]
2) For 2015, x = 15,
Hence, online jewelry sales for 2015 would be,
[tex]y=2(1+0.081)^{15}=6.43302745602\approx 6.4\text{ billion}[/tex]
The price of a sweatshirt at a local shop is twice the price of a pair of shorts.
The price of a T-shirt at the shop is $4 less than the price of a pair of shorts.
Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total
cost of $136.
1. Let w represent the price of one sweatshirt, t represent the price of one Tshirt, and h represent the price of one pair of shorts. Write a system of three
equations that represents the prices of the clothing.
2. Solve the system. Find the cost of each item.
Answer:
The cost of sweatshirt is 12$
Step-by-step explanation:
so we are going to make 3 equations and solve it using the substation method
First- We know the price of the shirt is two times more so w = 2h
Second we know that the the tee shirt is 4 dollars less then the pair of shoes so t = h − 4
Third we take the combined of Brads purchases 3w + 5t + 2h =136
w = 2h
t = h − 4
3w + 2h + 5t = 136
So to solve we are going to to the sub method
3(2h) + 2h + 5(h-4) = 136 - just re written, now we get rid of the ( )
6h + 2h + 5h -20 = 136 - Now we need to move the -20 and add it too 136
6h + 2h + 5h = 156 - Now sum up the H's
13h = 156 - Now divide 13 by 156
h= 12
so the Cost of a sweatshirt is 12 dollars
The cost of a sweatshirt is $24.
The cost of a t shirt is $8
The cost of shorts is $12
Step-by-step explanation:Let w represent the price of one sweatshirt
Let t represent the price of one Tshirt
Let h represent the price of one pair of shorts
The price of a sweatshirt at a local shop is twice the price of a pair of shorts.
[tex]w=2h[/tex] ...(1)
The price of a T-shirt at the shop is $4 less than the price of a pair of shorts.
[tex]t=h-4[/tex] ....(2)
Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.
[tex]3w+2h+5t=136[/tex] .....(3)
Substituting the values of w and t in (3)
[tex]3(2h)+2h+5(h-4)=136[/tex]
=> [tex]6h+2h+5h-20=136[/tex]
=> [tex]13h=136+20[/tex]
=> [tex]13h=156[/tex]
h = 12
t = h-4
[tex]t=12-4=8[/tex]
t = 8
w = 2h
[tex]w=2\times12=24[/tex]
w = 24
-----------------------------------------------------------------------------------
So, the cost of a sweatshirt is $24.
The cost of a t shirt is $8
The cost of shorts is $12
Water use in the summer is normally distributed with a mean of 311.4 million gallons per day and a standard deviation of 40 million gallons per day. City reservoirs have a combined storage capacity of 350 million gallons. The probability that a day requires more water than is stored in city reservoirs is P(X > 350)= 1 - P (Z < b). What is the value of b? Please report your answer in 3 decimal places.
Answer: 0.965
Step-by-step explanation:
Given : Water use in the summer is normally distributed with
[tex]\mu=311.4\text{ million gallons per day}[/tex]
[tex]\sigma=40 \text{ million gallons per day}[/tex]
Let X be the random variable that represents the quantity of water required on a particular day.
Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]
[tex]\dfrac{350-311.4}{40}=0.965[/tex]
Now, the probability that a day requires more water than is stored in city reservoirs is given by:-
[tex]P(x>350)=P(z>0.965)=1-P(z<0.965)[/tex]
We can see that on comparing the above value to the given P(X > 350)= 1 - P(Z < b) , we get the value of b is 0.965.
A recent report in a women magazine stated that the average age for women to marry in the United States is now 25 years of age, and that the standard deviation is assumed to be 3.2 years. A sample of 50 U.S. women is randomly selected. Find the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years
Answer: 0.0136
Step-by-step explanation:
Given : Mean : [tex]\mu=\ 25[/tex]
Standard deviation : [tex]\sigma= 3.2[/tex]
Sample size : [tex]n=50[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x = 24
[tex]z=\dfrac{24-25}{\dfrac{3.2}{\sqrt{50}}}=-2.20970869121\aprox-2.21[/tex]
The P-value =[tex]P(z\leq24)=0.0135526\approx0.0136[/tex]
Hence, the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years = 0.0136
Nadia has 20 more postcards than Pete. After Nadia gives Pete some postcards, Pete has 2 more postcards than Nadia. How many postcards does Nadia give to Pete? 2.
Answer:
11.
Step-by-step explanation:
N = Nadia.
P = Pete.
S = postcards that Nadia gives to Pete.
N = 20 + P
P + S = 2+N-S
To calculate S, we replaces N of the first equation in the second equation:
P + S = 2 + 20 + P-S
2S = 2 + 20 + P - P
2S = 22
S = 22/2 = 11.
Find the term of a loan of $225 at 3.5% if the simple interest is $63.
Answer: 8
Step-by-step explanation:
The formula to calculate the simple interest is given by :-
[tex]S.I. =Prt[/tex], where P is the principal amount , r is rate of interest and t is time.
Given: The principal amount : P = $225
The rate of interest : r = 3.5% =0.035
Simple Interest : SI = $63
Put these value in the above formula , we get
[tex] 63=225\times0.035t\\\\\Rightarrow\ t=\dfrac{63}{225\times0.035}\\\\\Rightarrow\ t=8[/tex]
Hence, the term of loan = 8