find the volume of the solid that results when the region bounded by y=x−−√, y=0 and x=36 is revolved about the line x=36.
The volume of the solid obtained by revolving the region bounded by y = x - √x, y = 0, and x = 36 around the line x = 36 can be found using the method of cylindrical shells. The resulting volume is approximately 3,012 cubic units.
To calculate the volume, we integrate the formula for the volume of a cylindrical shell, which is given by V = 2π∫[a,b] x * h(x) dx, where [a,b] represents the range of x values.
In this case, the lower bound of integration is 0 and the upper bound is 36, since the region is bounded by y = 0 and x = 36. The height of the cylindrical shell, h(x), is given by the difference between the x-coordinate of the curve y = x - √x and the line x = 36.
To obtain the x-coordinate of the curve, we set x - √x = 0 and solve for x. This gives us x = 0 or x = 1.
Next, we calculate the difference between x and 36, which gives us the height of the cylindrical shell. Then, we substitute the expressions for x and h(x) into the volume formula and integrate with respect to x.
After performing the integration, we find that the volume of the solid is approximately 3,012 cubic units.
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Cierra is buying juice. She needs 5 liters. A half liter juice cost $2.86. A 250-milliliter container of juice costs $1.05. What should Cierra buy so she gets 5 liters at the lowest price?
Answer: 250 mL Juice container
Step-by-step explanation:
Given
Half liter juice costs $2.86 i.e.
[tex]\dfrac{1}{2}\ L\rightarrow\$2.86\\\\1\ L\rightarrow\dfrac{2.86}{\frac{1}{2}}=\$5.72\\\\5\ L\rightarrow\$28.6[/tex]
A 250 mL juice costs $1.05 i.e.
[tex]250\ mL=0.25\ L\rightarrow \$1.05\\\\1\ L\rightarrow \dfrac{1.05}{0.25}=\$4.2\\\\\Rightarrow 5\ L\rightarrow \$21[/tex]
The cost of 250 mL Juice packet is low for 5 L quantity, therefore, Cierra must buy 250 mL Juice container
. **y" + xy' + y = 0, y(t) = 3 . y'(1)=4 (12pts) 3. Solve the Cauchy-Euler IVP:
The solution to the Cauchy-Euler initial value problem is -3/2
To solve the Cauchy-Euler initial value problem, we need to find the general solution of the differential equation and then use the initial conditions to determine the specific solution.
The given Cauchy-Euler differential equation is:
y" + xy' + y = 0
To solve this equation, we assume a solution of the form [tex]y(x) = x^r[/tex]
Differentiating twice with respect to x, we have:
[tex]y' = rx^{r-1}[/tex] and y" = [tex]r(r-1)x^{r-2}[/tex]
Substituting these expressions into the differential equation, we get:
[tex]r(r-1)x^{r-2} + x(rx^{r-1}) + x^r = 0[/tex]
[tex]r(r-1)x^{r-2} + r*x^r + x^r = 0[/tex]
[tex]x^{r-2}(r(r-1) + r + 1) = 0[/tex]
For a non-trivial solution, the expression in parentheses must equal zero:
r(r-1) + r + 1 = 0
Expanding and rearranging, we have:
[tex]r^2 - r + r + 1 = 0\\r^2 + 1 = 0[/tex]
The roots of this equation are complex numbers:
r = ±i
Therefore, the general solution of the Cauchy-Euler differential equation is:
[tex]y(x) = c_1x^i + c_2x^{-i}[/tex]
To simplify the solution, we can rewrite it using Euler's formula:
[tex]y(x) = c_1x^i + c_2x^{-i}\\ = c_1(cos(ln(x)) + i*sin(ln(x))) + c_2(cos(ln(x)) - i*sin(ln(x)))\\ = (c_1 + c_2)cos(ln(x)) + (c_1 - c_2)i*sin(ln(x))[/tex]
Now, let's apply the initial conditions to find the specific solution. We are given:
y(t) = 3 and y'(1) = 4
Substituting x = t into the solution, we have:
[tex](c_1 + c_2)cos(ln(t)) + (c_1 - c_2)i*sin(ln(t)) = 3[/tex]
To satisfy this equation, the real parts and imaginary parts on both sides must be equal.
From the real parts:
[tex](c_1 + c_2)cos(ln(t)) = 3[/tex]
From the imaginary parts:
[tex](c_1 - c_2)i*sin(ln(t)) = 0[/tex]
Since sin(ln(t)) ≠ 0 for any t, we must have ([tex]c_1 - c_2[/tex]) = 0.
This implies [tex]c_1 = c_2[/tex].
Substituting [tex]c_1 = c_2[/tex] into the real part equation, we get:
[tex]2c_1cos(ln(t)) = 3[/tex]
Solving for [tex]c_1[/tex], we find:
[tex]c_1 = 3/(2cos(ln(t)))[/tex]
Therefore, the specific solution of the Cauchy-Euler initial value problem is:
y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))
Now, we can find y'(1) by differentiating the specific solution with respect to x and evaluating it at x = 1:
y'(x) = -(3/2)(ln(t)sin(ln(x)) + cos(ln(x)))
y'(1) = -(3/2)(ln(t)sin(ln(1)) + cos(ln(1)))
= -(3/2)(ln(t)(0) + 1)
= -3/2
Therefore, the solution to the Cauchy-Euler initial value problem is:
y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))
y(t) = 3
y'(1) = -3/2
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Consider the region in the xy-plane bounded from above by the curve y=4x−x^2 and below by the curve y=x. Find the centroid of the region. (i.e. the center of mass of this region if the mass density is p =1)
The centroid of the region bounded from above by the curve y = 4x - x² and below by the curve y = x is (2/3, 4/3).
The region is bounded from above by the curve y = 4x - x² and below by the curve y = x. We need to find the points of intersection between these two curves. Setting the equations equal to each other,
4x - x² = x
Rearranging,
x² - 3x = 0
Factoring,
x(x - 3) = 0
So, x = 0 or x = 3.
The region is bounded from x = 0 to x = 3. To find the y-values within this region, we evaluate the equations y = 4x - x² and y = x at these x-values.
For x = 0,
y = 4(0) - (0)² = 0
For x = 3,
y = 4(3) - (3)² = 12 - 9 = 3
Thus, the y-values within the region are y = 0 to y = 3. Now, we calculate the area of the region by integrating the difference of the upper and lower curves,
A = ∫[0,3] [(4x - x²) - x] dx
A = ∫[0,3] (3x - x²) dx
A = [3x²/2 - x³/3] evaluated from x = 0 to x = 3
A = [27/2 - 9/3] - [0 - 0]
A = [27/2 - 3] - 0
A = 21/2
Now, for the centroid,
x = (1/A) * ∫[0,3] x * [(4x - x²) - x] dx
Simplifying,
x = (1/A) * ∫[0,3] (3x² - x³) dx
x = (1/A) * [x³ - x⁴/4] evaluated from x = 0 to x = 3
x = (1/A) * [(3)³ - (3)⁴/4] - [0 - 0]
x = (1/A) * [(27) - (81)/4] - 0
x = (1/A) * [(108 - 81)/4]
x = (1/A) * (27/4)
x = 27/(4A)
x = 27/(4 * 21/2)
x = 2/3, and,
x = (1/A) * ∫[0,3] [(4x - x²) - x]² dx
Simplifying,
y = (1/A) * ∫[0,3] (16x² - 8x³ + x⁴) dx
y = (1/A) * [(16x³/3 - 8x⁴/4 + x⁵/5)] evaluated from x = 0 to x = 3
y = (1/A) * [(16(3)³/3 - 8(3)⁴/4 + (3)⁵/5)] - [0 - 0]
y = (1/A) * [(16 * 27/3 - 8 * 81/4 + 243/5)]
y = (1/A) * [(144/3 - 648/4 + 243/5)]
y = (1/A) * [(480 - 972 + 243)/60]
y = (1/A) * (480 - 972 + 243)/60
y = -83/(20A)
Since A = 21/2, we can substitute it in,
y = -83/(20 * 21/2)
y = -83/(210/2)
y = -83/(105)
y = -4/5
Therefore, the centroid of the region is (2/3, 4/3).
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Use the Divergence Theorem to compute the net outward flux of the vector field F = (x², - y², z²) across the boundary of the region D, where D is the region in the first octant between the planes z = 9 - x - y and z = 6 - x - y.
To apply the Divergence Theorem, we need to first find the divergence of the vector field F:
div(F) = ∂/∂x(x²) + ∂/∂y(-y²) + ∂/∂z(z²)
= 2x - 2y + 2z
Next, we find the bounds for the region D by setting the two plane equations equal to each other and solving for z:
9 - x - y = 6 - x - y
z = 3
So the region D is bounded below by the xy-plane, above by the plane z = 3, and by the coordinate planes x = 0, y = 0, and z = 0. Therefore, we can set up the integral using the Divergence Theorem as follows:
∫∫F · dS = ∭div(F) dV
= ∭(2x - 2y + 2z) dV
= ∫₀³ ∫₀^(3-z) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx
We can simplify this integral using the limits of integration to get:
∫∫F · dS = ∫₀³ ∫₀^(3-x) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx
= ∫₀³ ∫₀^(3-x) [(2x - 2y)(3-x-y) + (2/3)(3-x-y)³] dy dx
= ∫₀³ [∫₀^(3-x) (2x - 2y)(3-x-y) dy + ∫₀^(3-x) (2/3)(3-x-y)³ dy] dx
Evaluating the two inner integrals, we get:
∫₀^(3-x) (2x - 2y)(3-x-y) dy = -x²(3-x) + (3/2)x(3-x)²
∫₀^(3-x) (2/3)(3-x-y)³ dy = (2/27)(3-x)⁴
Substituting these back into the integral and evaluating, we get:
∫∫F · dS = ∫₀³ [-x²(3-x) + (3/2)x(3-x)² + (2/27)(3-x)⁴] dx
= 9/5
Therefore, the net outward flux of the vector field F across the boundary of the region D is 9/5.
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please help with this?!?
Answer:
196.1
Step-by-step explanation:
Area of a circle is [tex]\pi r^{2}[/tex] so in order to find the radius you divide the diameter by 2 to get 7.9
Then you do [tex]7.9^{2}[/tex] x [tex]\pi[/tex] to get around 196.1
5 in = ___________ ft *Write your answers like this: whole number, one space, numerator, /, denominator. Example: 1 1/2 * PLEASE AWNSER FAST <3
Answer:
0.416667 ft
Step-by-step explanation:
plz help me and answer correctly for branliest
Answer:
It is complementary since their sum is equal to 90°
Please Help. What expression is equivalent to 6( t - 5 ) + 3
A. 6t - 2
B. 6t - 12
C. 3 ( 2t - 11 )
D. 3 ( 2t + 9 )
Use the normal distribution of SAT critical reading scores for which the mean is 505 and the standard deviation is 118. Assume the variable x is normally distributed. (a) What percent of the SAT verbal scores are less than 600? (b) If 1000 SAT verbal scores are randomly selected, about how many would you expect to be greater than 575? Click to view page 1 of the standard normal table. Click to view page 2 of the standard normal table. (a) Approximately 79 % of the SAT verbal scores are less than 600. (Round to two decimal places as needed.) (b) You would expect that approximately 722 SAT verbal scores would be greater than 575.
Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.
For a normal distribution of SAT critical reading scores with a mean of 505 and a standard deviation of 118, approximately 79% of the SAT verbal scores are less than 600. If 1000 SAT verbal scores are randomly selected, it is expected that approximately 722 of them would be greater than 575.
To determine the percentage of SAT verbal scores that are less than 600, we need to find the area under the normal distribution curve to the left of 600. We can use the standard normal distribution table or a statistical software to find the corresponding z-score.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
Substituting the values:
z = (600 - 505) / 118
z ≈ 0.8051
Using the standard normal distribution table, we can find the area to the left of z = 0.8051, which is approximately 0.7910.
To determine the percentage, we multiply the result by 100, giving us approximately 79% of SAT verbal scores that are less than 600.
For part (b), we can apply the same approach. We calculate the z-score for x = 575:
z = (575 - 505) / 118
z ≈ 0.5932
Using the standard normal distribution table, we find the area to the left of z = 0.5932, which is approximately 0.7242. This means that approximately 72.42% of SAT verbal scores are less than 575.
To estimate the number of SAT verbal scores greater than 575 in a sample of 1000, we multiply the percentage by the sample size:
Number of scores greater than 575 = 0.7242 * 1000 ≈ 722.
Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.
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Which expressions are equivalent to the one below? Check all that apply 5^x
Answer:
5 * 5^(x - 1) ; (15/3)^x ; 15^x / 3^x
Step-by-step explanation:
From the options, equivalent expressions include :
(15/3)^x
This is the same as ;
(15/3)^x
15 ÷ 3 = 5 ; then to the power of x = 5^x
15^x / 3^x ; since they are both raised to the same power, we can divide directly to obtain :
5^x
5 * 5^(x - 1)
5 = 5^1
5^1 * 5^(x-1)
5^(1 + x - 1) = 5^x
what is the approximate radius of a sphere with a volume of 900 cm squared
A 12 cm
B 36 cm
C 18cm
D 6cm
Answer:
about 5.99 or D. 6 cm
Step-by-step explanation:
you can use this formula
[tex]V=4/3 * \pi *r^{3}[/tex]
To test the hypothesis that the population standard deviation sigma-7.2, a sample size n=7 yields a sample standard deviation 5.985. Calculate the P- value and choose the correct conclusion. Your answer: The P-value 0.343 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.343 is The P-value 0.343 is significant and so strongly suggests that sigma<7.2. The P-value 0.192 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.192 is significant and so strongly suggests that sigma<7.2. The P-value 0.291 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.291 is significant and so strongly suggests that sigma<7.2. suggests that sigma<7.2. The P-value 0.309 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.309 is significant and so strongly suggests that sigma<7.2. The P-value 0.011 is not significant and so does not strongly suggest that sigma<7.2. The P-value 0.011 is significant and so strongly suggests that sigma<7.2.
The P-value of 0.343 is not significant and does not strongly suggest that the population standard deviation, sigma, is less than 7.2.
In hypothesis testing, the P-value is used to determine the strength of evidence against the null hypothesis. In this case, the null hypothesis is that the population standard deviation, sigma, is equal to 7.2. The alternative hypothesis is that sigma is less than 7.2.
To calculate the P-value, we need to compare the sample standard deviation, which is 5.985, to the hypothesized population standard deviation of 7.2. We can use the chi-square distribution to find the probability of observing a sample standard deviation as extreme as or more extreme than the one obtained, assuming the null hypothesis is true.
In this case, the P-value is 0.343. This means that if the null hypothesis is true, there is a 34.3% chance of obtaining a sample standard deviation of 5.985 or more extreme. Since the P-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. Therefore, we do not have strong evidence to suggest that the population standard deviation is less than 7.2.
In conclusion, the correct choice is: The P-value 0.343 is not significant and does not strongly suggest that sigma is less than 7.2.
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it always tells me i have to put 20 characters but i really need help
Answer:
B
Step-by-step explanation:
25/100 is 25%.
Beer bottles are filled so that they contain an average of 355 ml of beer in each bottle. Suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml. [You may find it useful to reference the z table.]
a. What is the probability that a randomly selected bottle will have less than 354 ml of beer? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
b. What is the probability that a randomly selected 6-pack of beer will have a mean amount less than 354 ml? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
c. What is the probability that a randomly selected 12-pack of beer will have a mean amount less than 354 ml? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
a. The probability that a randomly selected bottle will have less than 354 ml of beer is approximately 0.3085.
To calculate this probability, we convert the value of 354 ml to a z-score using the formula z = (x - μ) / σ, where x is the value we want to find the probability for (354 ml), μ is the mean (355 ml), and σ is the standard deviation (8 ml). By calculating the z-score, we can then look up the corresponding area under the normal distribution curve using a z-table. The z-score for 354 ml is approximately -0.125, and the corresponding area (probability) is 0.4508. Therefore, the probability of having less than 354 ml is 0.5 - 0.4508 = 0.0492 (or approximately 0.3085 when rounded to four decimal places).
b. The probability that a randomly selected 6-pack of beer will have a mean amount less than 354 ml is approximately 0.0194.
To calculate this probability, we need to consider the distribution of the sample mean. Since we are selecting a sample of size 6, the mean of the sample will have a standard deviation of σ / √n, where σ is the standard deviation of the population (8 ml) and n is the sample size (6). The standard deviation of the sample mean is therefore 8 ml / √6 ≈ 3.27 ml. We can then convert the value of 354 ml to a z-score using the same formula as in part a. The z-score for 354 ml is approximately -0.3061. By looking up this z-score in the z-table, we find the corresponding area (probability) of 0.3808. Therefore, the probability of the mean amount being less than 354 ml is 0.5 - 0.3808 = 0.1192 (or approximately 0.0194 when rounded to four decimal places).
c. The probability that a randomly selected 12-pack of beer will have a mean amount less than 354 ml is approximately 0.0022.
Similar to part b, we calculate the standard deviation of the sample mean for a sample size of 12, which is σ / √n = 8 ml / √12 ≈ 2.31 ml. By converting 354 ml to a z-score, we find a value of approximately -1.08. Looking up this z-score in the z-table, we find the corresponding area (probability) of 0.1401. Therefore, the probability of the mean amount being less than 354 ml is 0.5 - 0.1401 = 0.3599 (or approximately 0.0022 when rounded to four decimal places).
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PLEASEEEEEEEEEE HELPPPPPPPPPPPPP
Answer:
i dont kno bestie.. :/
Step-by-step explanation:
which statement best discribes the shape of the graph? the graph is skewed left. the graph is skewed right. the graph is nearly symmetrical. the graph is perfectly symmetrical.
The graph is nearly symmetrical.
Instead of using rigorous mathematics to solve this issue, let's simply look at it.
Most of the values are on the left side of a graph when it is skewed to the right.
The majority of values are on the right side of a graph when it is skewed left.
Perfect symmetry occurs when both sides are identical with regard to the median. Here, the means and medians are equal.
Nearly symmetrical would be very nearly perfect symmetry, with very minor variations on either side. Median and mean would be almost equal.
Now that we have counted the dots and have carefully examined them, we can rule out skewed right and skewed left. Is the graph now completely symmetrical? No!
Therefore, "nearly symmetrical" is the right response.
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Complete question =
The dot plot shows the number of words students spelled correctly on a pre-test. Which statement best describes the shape of the graph?
A.) The graph is skewed right.
B.) The graph is nearly symmetrical.
C.) The graph is skewed left.
D.) The graph is perfectly symmetrical.
A restaurant sells an 8-oz drink for $2.56 and a 12 oz drink for $3.66. Which drink is the better buy? i need help fast :(
Answer:
12 oz
Step-by-step explanation:
2.56 ÷ 8 = 0.32 per oz
3.66 ÷ 12= 0.305 per oz
PLSS HELP IMMEDIATELY!!! i’ll give brainiest if u don’t leave a link!
Answer: Evaluate the findings to compare to his hypothesis
Step-by-step explanation: Since the biologist already has the findings and has a hypothesis, he now has to compare both of them together.
Number 5 please helpppppppppp 10 points
Solve the system of equations.
5y - 4x = -7
2y + 4x = 14
X=
y =
Step-by-step explanation:
7y = 7
y = 1
2(1) + 4x = 14
4x = 12
x = 3
What is the answer to this question?
Verify the equation: (cos x + 1)/(sin^3 x) = (csc x)/(1 - cos x)
Answer:
dont know sorry
Step-by-step explanation:
what is the volume of each cylinder with a radius of 2.7 cm and a height of 5 cm
Answer:
114.51
Step-by-step explanation:
I'm not to sure what you meant by 'each' so I solved it like there was only one cylinder. hope this helped
Pls help and if you can show me how you do it :)
Find the number less than 40, that is
divisible by 5, and when divided by 6
has a remainder of 2.
what is 1/3 plus 1/2 in fraction form
Answer:
5/6
Step-by-step explanation:
Hope this helped!!!
If y varies directly as x, and y = 6 when x = 4, find y when x = 12.
y =
y=14 I hope this helps!!
Need help with all the question
Answer:
Step-by-step explanation:
So in ratios you can mostly all of the time scale your answer. So by determining how much increase there is in the baby's thigh bone each week you can pretty much answer these questions.
keep in mind: Proportional means having the same ratio. A scale factor is the ratio of the model measurement to the actual measurement in simplest form.
Example from https://www.mathsisfun.com/numbers/ratio.html
A ratio says how much of one thing there is compared to another thing.
ratio 3:1
There are 3 blue squares to 1 yellow square
Ratios can be shown in different ways:
Use the ":" to separate the values: 3 : 1
Or we can use the word "to": 3 to 1
Or write it like a fraction: 31
A ratio can be scaled up:
ratio 3:1 is also 6:2
Here the ratio is also 3 blue squares to 1 yellow square,
even though there are more squares.
An agronomist measures the lengths of n = 26 ears of corn. The mean length was 31.5 cm and the standard deviation was s= 5.8 cm. Find the Upper Boundary for a 95% confidence interval for mean length of corn ears. O 57.5 29.2 O 0.05 O 33.8
The upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm
To find the upper boundary for a 95% confidence interval for the mean length of corn ears, we can use the formula:
Upper Boundary = Mean + (Critical Value * Standard Error)
The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be obtained from the standard normal distribution, which is approximately 1.96.
The standard error is calculated by dividing the standard deviation by the square root of the sample size:
Standard Error = s / [tex]\sqrt{(n)}[/tex]
Given that the mean length was 31.5 cm (Mean) and the standard deviation was s = 5.8 cm, and the sample size was n = 26, we can calculate the upper boundary as follows:
Standard Error = 5.8 / [tex]\sqrt{26}[/tex] ≈ 1.138
Upper Boundary = 31.5 + (1.96 * 1.138) ≈ 33.8
Therefore, the upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm.
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PLEASE SOMEONE HELPPPPPPP
Answer:
12, 25, 26, 26, 26, 34, 35, 39, 42, 42, 50, 72.
Step-by-step explanation:
A stem and leaf plot works like a digit separator. The left is the first number, which is usually repeated, and the right is the number you add to it.
In this example, 3 is used three times for the numbers 34, 35, and 39.