Find a vector function r(t), that represents the curve of intersection of the two surfaces. the cylinder x2 y2=36 and the surface z=4xy
Answers
Given:
\(\begin{aligned}&x^2+y^2=16 \\&z=x y\end{aligned}\)
Express 16 as \(4^{2}\): \(x^2+y^2=16\)
\(x^2+y^2=4^2\\x^2+y^2=4^2 \times 1\)
Trignometry,
\(\cos ^2(t)+\sin ^2(t)=1\)
Now, substitute \(\cos ^2(t)+\sin ^2(t)\) for 1:
\(\begin{aligned}&x^2+y^2=4^2 \times 1 \\&x^2+y^2=4^2 \times\left[\cos ^2(t)+\sin ^2(t)\right]\end{aligned}\\x^2+y^2=4^2 \times \cos ^2(t)+4^2 \times \sin ^2(t)\)
Law of indicates:
\(\begin{aligned}&x^2+y^2=[4 \times \cos (t)]^2+[4 \times \sin (t)]^2 \\&x^2+y^2=[4 \cos (t)]^2+[4 \sin (t)]^2\end{aligned}\\x^2=[4 \cos (t)]^2 \text { and } y^2=[4 \sin (t)]^2\)
Taking positive square roots as follows:
\(x=4 \cos (t), y=4 \sin (t)\)
Recall that, z = xy.
Now, we have:
\(\begin{aligned}&z=4 \cos (t) \times 4 \sin (t) \\&z=16 \cos (t) \cdot \sin (t)\end{aligned}\)
Now, substitute the values:
\(r(t)=x_t i+y_t j+z_t k\)
So, the vector r(t) is: \(r(t)=(4 \cos (t)) i+(4 \sin (t)) i+(16 \cos (t) \cdot \sin (t)) i\)
Therefore, the vector function r(t) is written as: \(r(t)=x_t i+y_t j+z_t k\)
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Related Questions
write the equation of the line that contains the following points in standard form (16,-3),(-4,12)
Answers
Answer:
y = -3/4x + 9
Step-by-step explanation:
first, find the slope by taking the difference of the y-values / difference of the x-values
(-3-12) / (16-(-4)) which equals -15/20 or -3/4
now use that value to represent the 'm' in the formula in order to find 'b'
y = mx + b
-3 = -3/4(16) + b
-3 = -12 + b
9 = b
y = -3/4x + 9
Question 1 of 10
Which of the following is the converse of the statement "If it is snowing, then
it is my birthday"?
A. If it is my birthday, then it is not snowing.
B. If it is not my birthday, then it is snowing.
C. If it is not my birthday, then it is not snowing.
D. If it is my birthday, then it is snowing.
SUBT
Answers
Answer:
The converse of the statement "If it is snowing, then it is my birthday" is "If it is my birthday, then it is snowing." Therefore, the correct answer is D.
Warm-Up:
Solve using the distributive property.
1. 7 (3x + 8y-22)
2. -2 (4a + 3b-4)
Answers
Answer:
1. 7 (3x + 8y - 22) = 2 1 + 5 6 − 1 5 4
2. -2 (4a + 3b-4) = − 8 − 6 + 8
Step-by-step explanation:
Answer: Answer: 1. = 21x + 56y -154
2. -8a - 6b + 8
DE is a Mid-segment of triangle ABC. How long is DE?
Answers
Answer:
DE = 14
Step-by-step explanation:
Since, DE is a Mid-segment of triangle ABC.
Therefore,
DE = 1/2 * AC
2 DE = AC
2(5x - 1) = 3x + 19
10x - 2 = 3x + 19
10x - 3x = 19 +2
7x = 21
x = 21/7
x = 3
DE = 5x - 1
DE = 5*3 - 1
DE = 15 - 1
DE = 14
in the scientific method, which of the following is true about a hypothesis? group of answer choices the same hypothesis may not be tested more than once. research studies are designed to prove a hypothesis. a specific hypothesis is generated based on an established theory. a hypothesis both explains and predicts a phenomenon.
Answers
For the scientific method , the true statement about the hypothesis is (b) research studies are designed to prove a hypothesis.
The term Hypothesis is defined as the observation which is proposed as the possible outcome or results of an experiment .
we know that ; in order to prove a hypothesis, the experiments are designed and after that they are performed based on given hypothesis.
That is how the experiments are designed based on a previous idea which may result in a particular outcome.
So , any hypothesis is dependent on a specific experiment( research studies) .
Therefore , the correct option is (b) .
The given question is incomplete , the complete question is
In the scientific method, which of the following is true about a hypothesis?
(a) the same hypothesis may not be tested more than once.
(b) research studies are designed to prove a hypothesis.
(c) a specific hypothesis is generated based on an established theory.
(d) a hypothesis both explains and predicts a phenomenon.
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The right triangle on the right is a scaled copy of the right triangle on the left. Identify the scale factor. Express your answer in simplest form. ( wouldn't allow me to insert full picture. will message it to you.)
Answers
To find the scale factor, we just have to divide
\(\frac{8}{20}=\frac{2}{5}\)Hence, the scale factor is 2/5.Match the graph with the correct equation.
Answers
Answer:
work shown and pictured
Meghan has a jar containing 15 counters. There are only blue counters, green counters and red counters in the jar.
Hector is going to take at random one of the counters from his bag of 12 counters. He will look at the counter and put the counter back into the bag.
Hector is then going to take at random a second counter from his bag. He will look at the counter and put the counter back into the bag.
Meghan is then going to take at random one of the counters from her jar of counters. She will look at the counter and put the counter back into the jar.
The probability that the 3 counters each have a different colour is 7/24
(c) Work out how many blue counters there are in the jar.
Answers
The probability that the first two counters Hector draws are different colors is 1 - (the probability that the first two counters are the same color).
The probability that the first two counters Hector draws are the same color is:
(number of counters of that color in the bag / total number of counters in the bag) * (number of counters of that color in the bag - 1 / total number of counters in the bag - 1)
There are three cases to consider:
1. The first two counters are both blue:
(x/15) * ((x-1)/(15-1))
2. The first two counters are both green:
((15-x)/15) * ((15-x-1)/(15-1))
3. The first two counters are both red:
((15-x)/15) * ((15-x-1)/(15-1))
So the probability that the first two counters are the same color is:
(x/15) * ((x-1)/(14)) + ((15-x)/15) * ((14-x)/(14))
Simplifying:
= (x^2 - x + 210 - 15x^2 + 15x) / (15 * 14)
= (14x^2 + 210) / 210
= (2x^2 + 30) / 30
= (x^2 + 15) / 15
The probability that the three counters each have a different color is 7/24, so:
Probability (three different colors) = Probability (first two are different colors) * Probability (third is a different color)
= (1 - (x^2 + 15) / 15) * (x/15 + (15-x)/15 * x/14)
= [(225 - x^2 - 15) / 225] * [(15x + 210 - x^2) / 210]
= [(210 - x^2) / 225] * [(15x + 210 - x^2) / 210]
= (14 - x^2) / 15 * (3x + 42 - x^2) / 14
= (2 * 7 - x^2/15) * (3x + 42 - x^2) / 14
= (14 - x^2/15) * (3x + 42 - x^2) / 14 = 7/24
Multiplying both sides by 14:
(14 - x^2/15) * (3x + 42 - x^2) = 7/24 * 14
(14 - x^2/15) * (3x + 42 - x^2) = 49/12
Expanding:
42 - 3x^2/15 + 42x/15 - x^2 + x^3/15 = 49/12
Simplifying:
4x^3 - 36x^2 + 168x - 280 = 0
Dividing both sides by 4:
x^3 - 9x^2 + 42x - 70 = 0
By trying integer values for x, we find that x=5 is a solution to the equation.
Therefore, there are 5 blue counters in the jar.
Express 2x^2-12x-7 in the form a(x+b)^2+c
Answers
Answer:
2(x - 3)^2 - 25.
Step-by-step explanation:
2x^2-12x-7
= 2(x^2 - 6x) - 7
= 2[(x - 3)^2 - 9] - 7
= 2(x - 3)^2 - 18 - 7
= 2(x - 3)^2 - 25.
PLEASE HELP IM GONNA CRY LOL
An angle's measure is equal to 24 less than its complement. Find the measure of the angle.
Answers
Answer:
x = 33°
Step-by-step explanation:
x = angle
a complement of an angle =(90-x)
x = (90-x) - 24
x = 90 - x - 24
x = 66 -x
2x = 66
x = 33
Fiona has proved that a function, f(x), is an arithmetic sequence. How did she prove that?
Answers
Answer:
D. She showed that f(n) - f(n - 1) was a constant difference.
Step-by-step explanation:
which verbal expression represents the algebraic expression x/2+5
Answers
The verbal expressions A. half of five more than a number, C. five more than half a number, and D. half of five less than a number represent the given algebraic expression when assigned with a variable. The expressions are 1/2(x + 5), 5 + 1/2x, and 1/2(x - 5).
The verbal expressions that represent the algebraic expressions are A. half of five more than a number, C. five more than half a number, and D. half of five less than a number. To convert these expressions into algebraic form, we need to assign a variable, say x, to the unknown number.
A. Half of five more than a number can be expressed algebraically as 1/2(x + 5). B. Twice a number and five can be written algebraically as 2x + 5. C. Five more than half a number can be expressed algebraically as 5 + 1/2x. D. Half of five less than a number can be written algebraically as 1/2(x - 5).
Therefore, the expressions that represent the given algebraic expression are A. half of five more than a number, C. five more than half a number, and D. half of five less than a number. Expression B represents a different algebraic expression altogether.
To summarize, three of the given verbal expressions represent the given algebraic expression, which can be converted to algebraic form by assigning a variable to the unknown number. These expressions are 1/2(x + 5), 5 + 1/2x, and 1/2(x - 5).
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Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. Note the equations of the two cylinders could be the following x2+y2=r2,y2+z2=r2 which their axes are at right angles to each other. To solve the volume we will be using the triple integrals formula in rectangular coordinates which is V=∫∫∫dzdydx
Answers
The total volume of the two circular cylinders intersecting perpendicularly is\(V=πr2+πr3=πr2(1+r)\).
The volume of the two circular cylinders intersecting perpendicularly can be calculated using the triple integral formula in rectangular coordinates. The triple integral formula is V=∫∫∫dzdydx.
We can separate the integral into three components. The x-component is ∫dx from \(-√r2-y2 to √r2-y2\). The y-component is ∫dy from -r to r. The z-component is ∫dz from 0 to r.
Substituting the components into the equation for the triple integral, we get \(V=∫-rrdr∫-√r2-y2√r2-y2dx∫0rdz\).
We can solve the integral by splitting the y-component into two pieces:
\(V=∫-rrdr∫-r0dx∫0rdz+∫-rrdr∫0√r2-y2dx∫0rdz\)
The first integral can be solved easily and yields V=πr2. The second integral can be solved using the substitution u=r2-y2 and yields V=πr3.
Hence, the total volume of the two circular cylinders intersecting perpendicularly is V=πr2+πr3=πr2(1+r).
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Given the rule "if x is the nth term in Sequence A, x = 3n," can you use deductive reasoning to determine the next number in the pattern?
Answers
Answer:
yes; the next number is 3(n+1)
Step-by-step explanation:
Answer:
Yes, 3(n+1)
Step-by-step explanation:
Please listen to your teachers lessons lol.
Solve 19(x - 32)=?
Will mark brainlest
Answers
Answer:
19x - 608
Step-by-step explanation:
Answer: 19x - 608
Step-by-step explanation: Basically 32 x 19 (not the way teachers would want, but it still works -v-
Hope I helped!
please help me before tomorrow and explain with words please I don't understand :(
factor the trinomial
-2h^2+5h+3
Answers
2 km
Find the volume of the sphere. Show
your work.
Answers
Answer:
See answer below
Step-by-step explanation:
\(V = \frac{4}{3} \pi r^3\) 2km = radius
\(V=\frac{4}{3}\pi (2)^3\)
\(V = \frac{4}{3}\pi (8)\)
\(V = \frac{4}{3}(8)\pi\)
\(V =\frac{32}{3}\pi\) \(\pi =3.14\)
\(V = \frac{32}{3}(3.14)\)
\(V = 33.493\)
V≈ 33.49
Using π on the calculator
\(V = \frac{32}{3} \pi\)
\(V = 33.510\)
V ≈ 33.51
Please help me solve this
H(x)=2/3x-1
H(-3)
Answers
Answer:
-3
Step-by-step explanation:
So they are essentially giving you x, so you replace it with -3.
H(-3) = 2/3 (-3) -1
2/3 × -3 = -2
-2 - 1 = -3
H(-3) = -3 ⬅This is a true statement.
Hope this helps :)
divide: 8x^3 y^5 / 4x^2 y^3
Answers
\(2 {x}^{5} {y}^{8} \)
Step-by-step explanation:
yes
Carlos has a box of coins that he uses when playing poker
with friends. The box currently contains 56 coins,
consisting of pennies, dimes, and quarters. The number of
pennies is equal to the number of dimes, and the total
value is $6.59. How many of each denomination of coin
does he have?
Answers
Carlos has 28 pennies, 28 dimes, and 20 quarters.
What is value?Value is a concept that is used to describe the worth or importance of something or someone. It is often related to an individual's beliefs and opinions and is used to measure the quality of an item or service. Value is a subjective concept, as what is considered valuable to one person may not be valuable to another. Value can also be measured in terms of money, with items or services valued at a certain price.
The total value of the coins is $6.59. Each penny is worth 1 cent, each dime is worth 10 cents, and each quarter is worth 25 cents. Therefore, 28 pennies plus 280 dimes (28 x 10) plus 500 quarters (20 x 25) equals 858 cents, which is equal to $6.59.
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A vector
A
has components A
X
=83 m and A
y
=32 m. What is the magnitude of vector
A
?
Answers
The magnitude of vector A of the following components AX and AY is 88.95m
In this question we will apply the pythagoras theorem which will be depicted as
\(R = \sqrt{AX^{2} + AY^{2} }\) . . . . . . . . . (1)
where , R = resultant magnitude of vector
AX and AY are the components of vector
As per the question
AX = 83m
AY = 32m
Putting the values in the equation (1) we get
\(R= \sqrt{83^{2} +32^{2} }\)
\(R =\sqrt{6889+1024}\)
\(R=\sqrt{7913}\\\)
R = 88.95m
Thus the magnitude of vector A for the following components is 88.95m
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An international company has 25,200 employees in one country. If this represents 21.2% of the company's employees, how many employees does it have in total?
Round your answer to the nearest whole number.
Answers
Answer:
There are about 118868 employees at the company.
Step-by-step explanation:
When dealing with percentage problems, we can use the following equation:
P%x = y, where P% of x is y.
Since we're told that 25200 is 21.2% of the total number of employees, we substitute 21.2% for P and 25200 for y in the formula to solve for x, the total number of employees.
Furthermore, we must first convert the percentage to a decimal by dividing 21.2 by 100: 21.2 / 100 = 0.212
0.212x = 25200
Step 1: Divide both sides by 0.212 to solve for x:
(0.212x = 25200) / 0.212
x = 118867.9245
Step 2: Round to the nearest whole number to get the final answer.
118867.9245 rounded to the nearest whole number is 118868. Thus, the company has about 118867 employees in total.
Step 3: Check that 21.2% of 118868 is (exactly or approximately) 25200:
0.212 * 118868 = 25200
25200.016 > 25200
Although 25200.016 is slightly larger than 25200, we can still trust that our answer is 11868 since it's rounded and an approximation. A more exact number like 118867.9245 would give us exactly 25200.
Which aspect of Earth's orbital relationship to the Sun varies with a periodicity of both 400 Ka and 100 Ka?
Answers
These cycles are known as the eccentricity cycles, and they are one of the factors that contribute to the long-term climate variations on Earth
The aspect of Earth's orbital relationship to the Sun that varies with a periodicity of both 400 Ka and 100 Ka is the eccentricity of Earth's orbit. The eccentricity refers to the shape of Earth's orbit around the Sun, which is not a perfect circle, but an ellipse. The eccentricity of Earth's orbit changes over time due to gravitational interactions with other planets, particularly Jupiter and Saturn. When Earth's orbit is more elliptical, its distance from the Sun varies more throughout the year, leading to variations in climate and the amount of solar radiation received on Earth's surface. The periodicity of 400 Ka corresponds to a cycle of variations in Earth's eccentricity that affects the amount of solar radiation received at different latitudes and the distribution of ice ages. The periodicity of 100 Ka corresponds to a cycle of variations in Earth's eccentricity that affects the intensity of the seasons and the distribution of glacial periods. These cycles are known as the eccentricity cycles, and they are one of the factors that contribute to the long-term climate variations on Earth.
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Before changes to its management staff, an automobile assembly line operation had a scheduled mean completion time of 14.4 minutes. The standard deviation of completion times was 1.8 minutes. An analyst at the company suspects that, under new management, the mean completion time, u, is now less than 14.4 minutes. To test this claim, a random sample of 12 completion times under new management was taken by the analyst. The sample had a mean of 13.8 minutes. Assume that the population is normally distributed. Can we support, at the 0.05 level of significance, the claim that the population mean completion time under new management is less than 14.4 minutes? Assume that the population standard deviation of completion times has not changed under new management. Perform a one-tailed test.
a) State the null hypothesis H, and the alternative hypothesis.
b) Determine the type of test statistic to use.
c) Find the value of the test statistic. d) Find the p-value. e) Can we support the claim that the population mean completion time under new management is less than 14.4 minutes?
Answers
a) The null hypothesis (H0): The population mean completion time under new management is equal to or greater than 14.4 minutes. The alternative hypothesis (Ha): The population mean completion time under new management is less than 14.4 minutes. b) The type of test statistic to use is a one-sample z-test, since the sample size is small and the population standard deviation is known. c) The calculated test statistic is approximately -1.632. d) The p-value is slightly greater than 0.05. e) Based on the p-value being greater than the significance level (0.05), we fail to reject the null hypothesis.
a) The null hypothesis (H0): The population mean completion time under new management is equal to or greater than 14.4 minutes.
The alternative hypothesis (Ha): The population mean completion time under new management is less than 14.4 minutes.
b) Since the sample size is small (n = 12) and the population standard deviation is known, we will use a one-sample z-test.
c) The test statistic for a one-sample z-test is calculated using the formula:
z = (\(\bar x\) - μ) / (σ / √n), where \(\bar x\) is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values from the problem:
z = (13.8 - 14.4) / (1.8 / √12) ≈ -1.632
d) To find the p-value, we will compare the test statistic to the critical value from the standard normal distribution. At a significance level of 0.05 (α = 0.05), for a one-tailed test, the critical value is -1.645 (approximate).
The p-value is the probability of obtaining a test statistic more extreme than the observed test statistic (-1.632) under the null hypothesis. Since the test statistic is slightly larger than the critical value but still within the critical region, the p-value will be slightly greater than 0.05.
e) Since the p-value (probability) is greater than the significance level (0.05), we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the population mean completion time under new management is less than 14.4 minutes at the 0.05 level of significance.
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An aluminum bar 3 feet long weighs 12 pounds. What is the weight of a similar bar that is 3 feet 4 inches long?
Answers
Answer: 13 \(\frac{1}{3}\) pounds
Step-by-step explanation:
We will set up a proportion. I will have the weight, in pounds, on the top (numerator) and the length, in inches, on the bottom (denominator).
\(\frac{12}{36} =\frac{x}{40}\)
Cross-multiply:
36 * x = 12 * 40
36x = 480
Divide both sides of the equation by 36:
x = 13 \(\frac{1}{3}\) pounds
Answer:
Step-by-step explanation:
First find unit weight.
1 foot = 12 inches
3 ft = 3*12 = 36 inches
3ft 4 inches = 36 + 4 = 40inches
Weight of 36 inches long bar = 12 pounds
Weight of 1 inch bar = 12/36
\(\text{Weight of 40 inches long bar = $\dfrac{12}{36}*40$}\)
= 13.3 pounds
To which subset of the real number system does the number 1.5¯¯¯ belong?
Answers
The subset of the real number system that the number 1.5¯(repeating decimal) belong to is rational numbers.
What are rational numbers?
A rational number serves as a number that is expressed as the ratio of two integers, and the denominator of it will not be equal to zero.
Therefore, from 1.5¯ = 1.555555555 The subset of the real number system that the number 1.5¯(repeating decimal) belong to is rational numbers.
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Select the correct answer.
A line passes through (10,5) and (13,-4). Write the equation of the line in standard form.
A. 3x - y = 5
B. 3x - y = 25
C. 3x + y = 25
D. 3x + y = 35
Answers
Answer:
D.3x+y+35
Step-by-step explanation:
Just took test
Answer:
D.3x+y+35
Step-by-step explanation:
LOVE is a kite. LV and OE are diagonals. The segments DV = 9cm and LE = 15cm, and
Answers
The lengths of LV and OE are 15cm, and the lengths of LD and EV are 3√7 cm and 9/√7 cm, respectively.
Since LOVE is a kite, LV and OE are perpendicular bisectors of each other. Let the length of LD be x, and the length of EV be y. Then, we can use the Pythagorean theorem and the fact that the diagonals bisect each other to set up two equations:
x² + (LV/2)² = DV²/4
y² + (OE/2)² = LE²/4
Simplifying each equation and substituting the given values, we get:
x² + (LV/2)² = 81/4
y² + (OE/2)² = 225/4
We also know that the diagonals bisect each other, so we can set up another equation:
LV/2 + OE/2 = LO = VE
Substituting the given value for LE, we get:
LV/2 + OE/2 = 15
Solving this equation for one of the variables, we get:
LV = 30 - OE
Substituting this expression into the first equation above, we get:
x² + ((30 - OE)/2)² = 81/4
Simplifying and rearranging, we get:
OE² - 60OE + 675 = 0
Using the quadratic formula, we get:
OE = (60 ± √(3600 - 2700)) / 2
OE = 15 or 45
If OE = 15, then LV = 30 - 15 = 15, and we can solve for x and y:
x² + 7.5² = 81/4
y² + 7.5² = 225/4
Solving these equations, we get:
x = 3√7
y = 9/√7
If OE = 45, then LV = 30 - 45 = -15, which is impossible for a length. Therefore, the solution is:
LV = OE = 15
x = 3√7
y = 9/√7
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Complete question:
LOVE is a kite. LV and OE are diagonals. The segments DV = 9cm and LE = 15cm are given. Find the lengths of the other segments of the diagonals, DV, OE, and LV.
Question 1(Multiple Choice Worth 2 points)
(Converting Between Systems MC)
I need help pleaseeee
For a craft project you need 182 inches of ribbon, but it is only sold by the meter. Determine the amount of ribbon, in meters, you need to buy for the project. (1 inch = 2.54 centimeters and 1 centimeter = 0.01 meter)
462
47
12
5
Answers
The amount of ribbon needed to buy for the project in meters is (d) 5.
According to question,
1 centimeter = 0.01 meter
2.54 centimeter = 0.01*2.54
= 0.0254 meters
1 inches = 2.54 centimeters
= 0.0254 meters
Amount of ribbon needed for project = 182 inches
= 182 * 0.0254 meters
= 4.62 meters
≅ 5 meters
Hence, 5 meters of ribbon is needed for the project.
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8) A dress normally sells for $35.85. How much does the dress cost after a 15% discount??
Answers
Answer:
$30.47
Step-by-step explanation:
100%-15%=85%
85%=0.85
$35.85*0.85=30.47 (to 2d.p)
Hope this helps