The maximum value of f on the ellipse is approximately 1.779 and the minimum value is approximately -1.779.
To find the maximum and minimum values of f(x, y) = 3x + y on the ellipse x^2 + 4y^2 = 1, we can use the method of Lagrange multipliers.
First, we define the Lagrangian function as L(x, y, λ) = 3x + y - λ(x^2 + 4y^2 - 1). We then find the partial derivatives of L with respect to x, y, and λ and set them equal to zero:
∂L/∂x = 3 - 2λx = 0
∂L/∂y = 1 - 8λy = 0
∂L/∂λ = x^2 + 4y^2 - 1 = 0
Solving these equations simultaneously, we obtain the critical points (±1/3√5, ±1/√20). We can then evaluate f at these critical points to find the maximum and minimum values:
f(1/3√5, 1/√20) ≈ 0.593
f(1/3√5, -1/√20) ≈ -0.593
f(-1/3√5, 1/√20) ≈ 1.779
f(-1/3√5, -1/√20) ≈ -1.779
Intuitively, the Lagrange multiplier method allows us to optimize a function subject to a constraint, which in this case is the ellipse x^2 + 4y^2 = 1.
The critical points of the Lagrangian function are the points where the gradient of the function is parallel to the gradient of the constraint, which correspond to the maximum and minimum values of the function on the ellipse.
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A bag of sweets contains only gobstoppers and sherbert lemons.
There are 3 gobstoppers for every 4 sherbert lemons.
There are 56 sweets in the bag. How many gobstoppers are there?
Four buses carrying 150 football fans from the same school arrive at a football stadium. The buses carry, respectively, 20, 45, 35, and 50 students. One of the fans is randomly selected. Let X denote the number of fans that were on the bus carrying the randomly selected person. One of the 4 bus drivers is also randomly selected. Let Y denote the fans of students on his bus. Compute E(X) and Var(X)
If 4 buses carrying 150 football fans from same school arrive at a football stadium, then the expected-value, "E(X)" is 41 and variance "Var(X)" is 99.
To find the expected value of X, we use the formula E(X) = ∑x P(X=x), where x is = possible values of X and P(X=x) = probability of X taking the value x.
Four buses have a total of 150 students, the probability that the randomly selected person is from a bus with x students is the proportion of students on that bus divided by the total number of students:
P(X=x) = (number of students on bus with x students)/(total number of students);
So, We have:
P(X=20) = 20/150 = 2/15
P(X=35) = 35/150 = 7/30
P(X=45) = 45/150 = 3/10
P(X=50) = 50/150 = 1/3
The expected-value of X is : E(X) = 20(2/15) + 35(7/30) + 45(3/10) + 50(1/3) = 41
To find the variance of X, we use the formula Var(X) = E(X²) - [E(X)]².
We already know E(X), so we need to find E(X²).
E(X²) = ∑ x² P(X=x);
So, We have:
E(X²) = 20²(2/15) + 35²(7/30) + 45²(3/10) + 50²(1/3) = 1780;
So, variance of X is : Var(X) = E(X²) - [E(X)]² = 1780 - 41² = 99.
Therefore, the expected value of X is 41 and the variance of X is 99.
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Just the answer is fine:)
Let S be the surface in R3 that lies on C = {(x, y, z) ER3 | 22 = 100(x2 + y²)} - and between the planes given by z= 1 and 2 = 5. Then the area of Sis = A(S) Check
The area of S is:
[tex]A(S) = 16\pi \sqrt{(500/11)}= 128.8[/tex]
How to find the area of S?The surface S can be described in terms of cylindrical coordinates by setting:
x = r cos(θ)
y = r sin(θ)
z = z
Using these coordinates, we can rewrite the equation for C as:
r² = 22/100(x² + y²) = 22/100r²
Simplifying this equation, we get:
[tex]r = \sqrt{(500/11)}[/tex]
Thus, the surface S is the portion of the cylinder of radius [tex]\sqrt{(500/11)}[/tex] between z = 1 and z = 5.
To calculate the area of S, we can use the formula:
A(S) = ∫∫∂S ||n|| [tex]dA[/tex]
where ||n|| is the magnitude of the normal vector to the surface, and [tex]dA[/tex] is the area element on the surface.
For the cylinder, the normal vector is simply the radial unit vector pointing outward from the origin:
n = (cos(θ), sin(θ), 0)
The magnitude of the normal vector is ||n|| = 1, so we can simplify the formula for the area to:
A(S) = ∫∫∂S [tex]dA[/tex]
To evaluate this integral, we need to parameterize the surface S. We can use the cylindrical coordinates we defined earlier:
x = r cos(θ)
y = r sin(θ)
z = z
with 0 ≤ θ ≤ 2π and 1 ≤ z ≤ 5.
The area element in cylindrical coordinates is given by:
[tex]dA = r \ dz\ d\theta[/tex]
Substituting in our parameterization of S, we get:
A(S) = ∫∫∂S r [tex]dz[/tex] dθ
[tex]= \int\limits^{2\pi }_0 \int\limits^5_1 {\sqrt{(500/11)} dz d\theta}\\= \sqrt{(500/11)} \int\limits^{2\pi }_0 {(5 - 1) d\theta}\\= 16\pi \sqrt{(500/11)[/tex]
Therefore, the area of S is:
[tex]A(S) = 16\pi \sqrt{(500/11)}= 128.8[/tex]
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The area of a rectangle is 42 square meters. The length is 7 meters. What is the width?
35 meters
14 meters
15 meters
6 meters
Answer: 6
Step-by-step explanation:42/7=6
the circumstances of the base is 48π cm. if the volume if the cone is 8640π cm cubed, what is the height?
Answer:
h = 15 cm
Step-by-step explanation:
Given:
C (base's circumstance) = 48π cm
V (volume) = 8640π cm^3
Find: h (height) - ?
[tex]c = 2\pi \times r[/tex]
[tex]2\pi \times r = 48π[/tex]
[tex]r = 24[/tex]
We found the radius
[tex]v = \frac{1}{3} \times \pi {r}^{2} \times h[/tex]
[tex] \frac{1}{3} \times \pi \times {24}^{2} \times h = 8640π[/tex]
Multiply the whole equation by 3 to eliminate the fraction:
[tex]1728\pi \times h = 25920\pi[/tex]
[tex]h = 15[/tex]
A sheep rancher plans to fence a rectangular pasture next to an irrigation canal. No fence will be needed along the canal, but the other three sides must be fenced. The pasture must have an area of 180,000 m² to provide enough grass for the sheep. Find the dimensions of the pasture which require the least amount of fence.
The dimensions of the pasture that require the least amount of fence are approximately 600 meters by 300 meters.
To minimize the amount of fence needed, we want to maximize the length of the side next to the canal. Let's call this side x and the other two sides y.
We know that the area of the rectangle must be 180,000 m², so we have x*y = 180,000. We want to minimize the amount of fence, which is the perimeter of the rectangle: P = x + 2y
To solve for the dimensions that require the least amount of fence, we need to eliminate one variable. We can do this by using the area equation to solve for one variable in terms of the other:
y = 180,000/x
Substituting this into the perimeter equation, we have:
[tex]P = x + 2(180,000/x)[/tex]
To find the minimum value of P, we take the derivative with respect to x and set it equal to zero:
[tex]P' = 1 - 360,000/x^2 = 0x = sqrt(360,000) ≈ 600[/tex]
Substituting this back into the area equation, we find:
[tex]y = 180,000/x ≈ 180,000/600 ≈ 300[/tex]
So, the dimensions of the pasture which require the least amount of fence are approximately 600 meters by 300 meters.
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The triangle above has the following measures.
q=8 in
m/Q = 37°
Find the length of sider.
Round to the nearest tenth and include correct units.
The triangle above has the following measures. The length of sider is 13.3 inches.
q = 8 inches
m ∠Q = 37°
sin (Q) = q/r
r = q / sin(Q)
= 8 / sin (37°)
= 13.3 inches
In Math, a triangle is a three-sided polygon that comprises of three edges and three vertices. The main property of a triangle is that the amount of the inward points of a triangle is equivalent to 180 degrees. This property is called point total property of triangle.
There are three points in a triangle. These points are framed by different sides of the triangle, which meets at a typical point, known as the vertex. The amount of every one of the three inside points is equivalent to 180 degrees.
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How did Americans lower their dependency on oil from the Middle East during the oil crisis?(1 point) A. They increased speed limits on highways. B. They increased speed limits on highways. C. They started to produce more of their own oil. They started to produce more of their own oil. They decreased the price of oil and gas by four times. They decreased the price of oil and gas by four times. They let people buy as much gas as they wanted. They let people buy as much gas as they wanted
During the oil crisis, Americans took various measures to reduce their dependency on oil from the Middle East. One of the key steps they took was to increase fuel efficiency standards for cars and trucks.
This helped to reduce the amount of oil needed to power vehicles. Additionally, they started to produce more of their own oil by opening up new oil fields and investing in alternative energy sources such as wind and solar power.
They also implemented policies to encourage conservation and reduce wasteful energy consumption. However, they did not decrease the price of oil and gas by four times, nor did they allow people to buy as much gas as they wanted.
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Please help and show work pls
noah and emma are standing on opposite sides of a 43.3 ft tree looking up at noah’s cat, which is perched at the very top. they are separated by a horizontal distance of 100 ft. the distance from emma to the cat is 50 ft and her angle of elevation to see the cat is 60°. the distance from noah to the cat is 86.6 ft and his angle of elevation to see the cat is 30°.
(a) identify three different trigonometric ratios that could have been used to find the distance x between emma and the base of the tree. for each trigonometric ratio, determine the distance. round to the nearest whole number.
(b) use the pythagorean theorem to find the distance x between emma and the base of the tree. round to the nearest whole number.
(a) The three different trigonometric ratios that could have been used to find the distance x between Emma and the base of the tree are sin(60°) = 43.3/50, cos(60°) = x / 50, and tan(60°) = 43.3 / x. The distance is 25 ft.
(b) Using the Pythagorean theorem, the distance x between Emma and the base of the tree is 25 ft.
(a) We can use sine, cosine, and tangent trigonometric ratios to find the distance x between Emma and the base of the tree.
1. Sine:
sin(60°) = opposite/hypotenuse = (tree height) / 50
tree height = 50 * sin(60°) = 43.3 ft (since it's given that the tree is 43.3 ft tall)
2. Cosine:
cos(60°) = adjacent/hypotenuse = x / 50
x = 50 * cos(60°) = 25 ft
3. Tangent:
tan(60°) = opposite/adjacent = (tree height) / x
x = (tree height) / tan(60°) = 43.3 / tan(60°) ≈ 25 ft
(b) To find the distance x between Emma and the base of the tree using the Pythagorean theorem, we can consider the triangle formed by Emma, the base of the tree, and the top of the tree.
Let's call the distance from the base of the tree to the top of the tree (tree height) y.
Emma's distance to the cat (50 ft) is the hypotenuse, the distance x is one leg, and the tree height y (43.3 ft) is the other leg of the right triangle.
Using the Pythagorean theorem: x² + y² = hypotenuse²
x² + 43.3² = 50²
x² + 1874.89 = 2500
x² = 625.11
x ≈ 25 ft
So, the distance between Emma and the base of the tree is approximately 25 ft.
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The table shown below gives the approximate distance from the sun for a few different planets how much further (in km) from the sun is Saturn than Venus
The distance that shows how much farther (in km) from the sun is Saturn than Venus is: [tex]1.2 * 10^9[/tex] km.
How to calculate the distanceAccording to the table, the distance of Saturn from the Sun is [tex]1.4 * 10^{9}[/tex] and the distance of Venus from the Sun is [tex]1.1 * 10^{8}[/tex] .
Now to determine how much further from the Sun is Saturn than Venus, we will subtract the distance of the planet with the higher distance span from the one with the lower distance.
So our calculation will go thus:
[tex]1.4 * 10^9 - 1.1 * 10^8 = \\140000000 - 11000000 = 1290000000\\= 1.29 * 10^9[/tex]
From the calculation above, we can see how much further from the sun, is Saturn than Venus.
Complete Question:
The table shown below gives the approximate distance from the sun for a few different planets. How much farther (in km) from the sun is Saturn than Venus? Express your answer in scientific notation.
_______km
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NEXT QUESTION >
Triangle HNR is shown where point K is the
centroid, KW = (2y — 8.9), KH
(2y 8.9), KH = (4.5w - 5.9),
KR = (0.5y + 3.2), KN = (5x – 5.2), KD = (9w
and KT = (7.1x – 11.8).
Z
W
H
K
Answer:
a b e
Step-by-step explanation:
If the probability that the Islanders will beat the Rangers in a game is 0.88, what is the probability that the Islanders will win at least three out of four games in a series against the Rangers? Round your answer to the nearest thousandth.
On Thursday,30 scholars went to morning homework help. On Friday, 24 scholars went. What is the percent decrease in the number of scholars who went to morning homework help from Thursday to Friday?
PLEASE HELP 20 POINTS
There is 20% decrease in scholars number who went to the morning homework help.
What is the percent decrease?To get the scholar's percent decrease, we need to calculate the difference between them on Thursday and Friday and theb divide that by the number of scholars on Thursday.
Data:
Number of scholars on Thursday = 30
Number of scholars on Friday = 24
Difference = 30 - 24 = 6
The percent decrease = (6/30) x 100%
The percent decrease = 0.2 * 100%
The percent decrease = 20%
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1.6.
horseshoe falls is one of the three waterfalls that make up niagara falls near buffalo,
new york. it has an average flow of 7 x 103 cubic meters per second. which value best
represents how much water goes over horseshoe falls hourly?
a. 1.94 x 101 cubic meters per hour
b.
4.20 x 105 cubic meters per hour
c.
2.52 x 10 cubic meters per hour
d.
6.05 x 108 cubic meters per hour
Horseshoe Falls has a flow rate of approximately 2.52 x 10⁷ cubic meters of water per hour, but the given options do not accurately represent this value.
Horseshoe Falls is one of the three waterfalls that make up Niagara Falls near Buffalo, New York. It has an average flow of 7 x 10³ cubic meters per second. To determine the amount of water that goes over Horseshoe Falls hourly, we need to convert the flow rate from cubic meters per second to cubic meters per hour.
There are 3600 seconds in an hour. To convert the flow rate to cubic meters per hour, we simply multiply the flow rate by the number of seconds in an hour:
(7 x 10³ cubic meters/second) x (3600 seconds/hour) = 25.2 x 10⁶ cubic meters/hour
The closest value to this among the given options is:
b. 4.20 x 10⁵ cubic meters per hour
However, there seems to be a typo or error in the provided options. The correct answer should be:
25.2 x 10⁶ cubic meters per hour (approximately 2.52 x 10⁷ cubic meters per hour)
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x= 3y-5 make y the subject
Answer:
y = (x + 5)/3
Step-by-step explanation:
To make y the subject, you need to isolate y on one side of the equation.
x = 3y - 5
Add 5 to both sides:
x + 5 = 3y
Divide both sides by 3:
y = (x + 5)/3
Therefore, y is the subject of the formula when it is expressed as:
y = (x + 5)/3
If x = 3 centimeters, y = 5 centimeters, and z = 5 centimeters, what is the area of the object?
A. 40 square centimeters
B. 50 square centimeters
C. 45 square centimeters
D. 25 square centimeters
The area of the object is 40 square centimeters
The correct answer is an option (B)
We know that the formula for the area of trapezoid is,
A = ((a + b)/2) × h
where a and b are the two parallel bases
h is the height of the trapezoid
From the attached figure, first we determine the length of the two parallel bases.
Let us assume that 'a' represents the length of the upper base and 'b' represents the length of the bottom side of the trapezoid
We can observe that a = 2x
so the length of a = 6 centimeters
And b = 2z
So, the length of b = 2(5)
= 10 cm
Here, the height h of the trapezoid is given by y = 5 cm
Using above formula for the area of trapezoid, the area of the object would be,
A = ((a + b)/2) × h
A = ((6 + 10)/2) × 5
A = (16/2) × 5
A = 40 cm²
Therefore, the correct answer is an option (B)
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Find the complete question below.
The heights of 14 plants, in inches, are listed.
12, 14, 15, 15, 16, 16, 16, 17, 18, 18 ,19, 20, 22, 25
If another plant with a height of 13 inches is added to the data, how would the mean be impacted?
The mean would stay the same value of about 17.1 inches.
The mean would decrease in value to about 17.1 inches.
The mean would stay the same value of about 17.4 inches.
The mean would increase in value to about 17.4 inches.
Answer:
The mean would decrease in value to about 17.1 inches.
Step-by-step explanation:
If we add a smaller data to the mean of a numerical data set, the mean will decrease.
Hope this helps.
Choose the equations that are equivalent. Select all that apply. A. 52 = 8n + 4 B. 4(2n + 1) = 52 C. 4 = 52 – 8n D. 4n = 48
A. 52 = 8n + 4, B. 4(2n + 1) = 52 and C. 4 = 52 – 8n are the equivalent equations.
Choosing the equations that are equivalentSimplifying the equations, we have
A. 52 = 8n + 4
8n = 48
n = 6
To see this, first simplify equation B:
4(2n + 1) = 52
8n + 4 = 52
8n = 48
n = 6
C. 4 = 52 – 8n
8n = 48
n = 6
Then simplify equation D:
4n = 48
n = 12
As you can see, equations A, B and C are equivalent and both simplify to n = 6.
Therefore, the correct answers are A, B and C.
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Use the Generalized Power Rule to find the derivative of the function.
f(x) = (3x + 1)^5(3x - 1)^6
This is the derivative of the given function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule.
To find the derivative of the function f(x) = (3x + 1)^5(3x - 1)^6 using the Generalized Power Rule, we will need to apply both the Product Rule and the Chain Rule.
The Product Rule states that if you have a function f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x).
First, let's identify g(x) and h(x) in your function:
g(x) = (3x + 1)^5
h(x) = (3x - 1)^6
Next, we'll find the derivatives g'(x) and h'(x) using the Chain Rule, which states that if you have a function y = [u(x)]^n, then y' = n[u(x)]^(n-1) * u'(x).
For g'(x):
u(x) = 3x + 1
n = 5
u'(x) = 3
g'(x) = 5(3x + 1)^(5-1) * 3 = 15(3x + 1)^4
For h'(x):
u(x) = 3x - 1
n = 6
u'(x) = 3
h'(x) = 6(3x - 1)^(6-1) * 3 = 18(3x - 1)^5
Now, we apply the Product Rule:
f'(x) = g'(x)h(x) + g(x)h'(x) = 15(3x + 1)^4(3x - 1)^6 + (3x + 1)^5 * 18(3x - 1)^5
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The critical number of the function f(x) = 5x2 + 7x – 10 is The function f(x) = -2x3 + 39x2 – 240x + 2 has two critical numbers A < B with A = and B =
The critical numbers of f(x) = -2x^3 + 39x^2 - 240x + 2 are A = 5 and B = 8.
To find the critical numbers of a function, we need to find the values of x where the derivative of the function is zero or undefined.
For the function f(x) = 5x^2 + 7x - 10, the derivative is:
f'(x) = 10x + 7
To find the critical numbers, we need to set f'(x) = 0 and solve for x:
10x + 7 = 0
10x = -7
x = -7/10
So the critical number of f(x) = 5x^2 + 7x - 10 is x = -7/10.
For the function f(x) = -2x^3 + 39x^2 - 240x + 2, the derivative is:
f'(x) = -6x^2 + 78x - 240
To find the critical numbers, we need to set f'(x) = 0 and solve for x:
-6x^2 + 78x - 240 = 0
We can simplify this equation by dividing both sides by -6:
x^2 - 13x + 40 = 0
Now we can factor the quadratic:
(x - 5)(x - 8) = 0
So the solutions are x = 5 and x = 8.
To determine which critical point is A and which is B, we need to check the sign of the second derivative of f(x) at each critical point.
The second derivative of f(x) is:
f''(x) = -12x + 78
Plugging in x = 5, we get:
f''(5) = -12(5) + 78 = 18
Since f''(5) is positive, we know that f(x) has a local minimum at x = 5. Therefore, x = 5 is the critical point A.
Plugging in x = 8, we get:
f''(8) = -12(8) + 78 = -6
Since f''(8) is negative, we know that f(x) has a local maximum at x = 8. Therefore, x = 8 is the critical point B.
Therefore, the critical numbers of f(x) = -2x^3 + 39x^2 - 240x + 2 are A = 5 and B = 8.
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A recent report states that 55% of U. S. Adults use Netflix to stream shows and movies. An advertising company believes the proportion of California residents who use Netflix is greater than the national proportion, because Netflix headquarters is located in Los Gatos, California. The company selects a random sample of 600 adults from California and finds that 360 of them use Netflix. Is there convincing evidence at the level that more than 55% of California residents use Netflix?
Calculated test statistic of 2.08 is greater than the critical value of 1.645, we reject the null hypothesis and conclude that there is convincing evidence at the 0.05 level that more than 55% of California residents use Netflix.
We can use a hypothesis testing approach to answer this question. The null hypothesis is that the true proportion of California residents who use Netflix is the same as the national proportion, or p = 0.55. The alternative hypothesis is that the true proportion of California residents who use Netflix is greater than 0.55, or p > 0.55.
We can use the sample proportion of Netflix users in California, which is 360/600 = 0.6, as an estimate of the true proportion p. The standard error of the sample proportion is:
SE = √[(p*(1-p))/n] = √[(0.55*(1-0.55))/600] = 0.024
The test statistic is:
z = (p - 0.55)/SE = (0.6 - 0.55)/0.024 = 2.08
Assuming a significance level of 0.05 and a one-tailed test (since the alternative hypothesis is one-sided), the critical z-value is 1.645.
Since our calculated test statistic of 2.08 is greater than the critical value of 1.645, we reject the null hypothesis and conclude that there is convincing evidence at the 0.05 level that more than 55% of California residents use Netflix. However, we should keep in mind that this conclusion is based on a sample of 600 adults from California, and there is always some degree of uncertainty involved in statistical inference based on samples.
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A teacher asked three different students to write the conditions that would result in a triangle. Which of the following students listed conditions that would result in more than one triangle?
The condition that will result in more than one triangle is C. Student III.
How the conditions will result in more than one triangleThe conditions listed by the third student will result in more than one triangle because we are given all three angles. As a rule in math, some conditions will determine if there is more than one triangle. One of them is this:
Rule 1:
If all three angles of the triangle are given and they all add up to exactly 180°, it is possible to get more than one triangle. In the third option, we are given angles 62°, 36°, and 82°, so different triangles can be constructed. Also, they all add up to give 180°. The condition is satisfied.
Rule 2:
Also, as a rule, if we have two angles that do not add up to 180° and one side, then only one unique triangle can be obtained. This is the case for student A who is given angles A and B and a side length of 5cm. (ASA)
Rule 3:
Student 2 will also produce a unique triangle because there are three sides that meet the triangle inequality theorem. (SSS)
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Dans une boite il ya 12 boules vertes et 6 boules bleues quelle est la proportion de boules vertes dans cette boite
La proportion de boules vertes dans cette boîte est de 2/3.
How to calculate the proportion of green balls in the box?Pour déterminer la proportion de boules vertes dans cette boîte, nous devons comparer le nombre de boules vertes au nombre total de boules dans la boîte.
Le nombre total de boules dans la boîte est la somme des boules vertes et des boules bleues, soit 12 + 6 = 18 boules.
Maintenant, pour calculer la proportion de boules vertes, nous divisons le nombre de boules vertes par le nombre total de boules.
Proportion de boules vertes = Nombre de boules vertes / Nombre total de boules
Proportion de boules vertes = 12 / 18
Simplifiant cette fraction, nous obtenons :
Proportion de boules vertes = 2/3
La proportion de boules vertes dans cette boîte est donc de 2/3 ou environ 66.67%.
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An archeologist finds part of a circular plate. What was the diameter of the plate? Justify your answer
A circular plate fragment is found by an archaeologist. The plate is 13.9 inches in diameter.
What are Chords?A chord in mathematics is a piece of a straight line that joins two points on a curve. A chord is a line segment with its endpoints on the curve, to be more precise.
The word "chord" is most frequently used in relation to the geometry of circles, where a chord is a line segment that joins two points on a circle's circumference. Given the lengths of the circle's radii and the separation between the chord's ends, the Pythagorean theorem can be used to determine the chord's length in this situation.
The relationship between two notes in music can also be represented by chords. A chord is, in this context, a grouping of three or more notes performed simultaneously to produce a harmonic sound. The structure of musical compositions can be analysed and understood using the mathematical concepts of chord progressions.
The equidistant chords theorem states that two chords are congruent in the same circle or a congruent circle if and only if they are equidistant from the centre.
Additionally, as depicted in the illustration, the supplied chords are equally spaced apart; as a result, they must meet in the circle's centre.
LHE is formed by connecting the points L and E to make a right-angled triangle ΔLHE.
The perpendicular chord bisector theorem states that if a circle's diameter is perpendicular to a chord, the diameter will also bisect the chord's arc HE≅ HD.
and IF≅IG
hence , HE=7/2
HD=7/2
IF=7/2
And IG=7/2
Applying the Pythagorean theorem, simplifying by addition, and substituting 6 for the perpendicular P, LE for the hypotenuse H, and 7/2 for the base B in the equation P²+B²=H².
P²+B²=H²
6²+7/2²=LE²
LE²= 36+ 12.25
LE²= 48.25
Since LE represents the radius of a circle, it is impossible for it to be negative, hence the negative value LE=6.94 is disregarded.
Since LE is the circle's radius, the circle's radius is 6.94. As a result, the circle has a 13.88 diameter.
The diameter should be rounded out to the closest tenth.
d≈13.9.
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Of the following, which option or options would help make this graph less misleading? i. the scale on the x-axis should be resized. ii. the scale on the y-axis should be resized. iii. the identity of the two parks should be more clearly differentiated. a. i and ii b. ii and iii c. iii only d. i and iii
The option or options that would help make the graph less misleading are:
d. i and iii. The scale on the x-axis should be resized. The identity of the two parks should be more clearly differentiated.
Resizing the scale on the x-axis (option i) would help provide a clearer picture of the difference between the two parks, as it would make it easier to see the differences in the number of visitors between the two parks.
Differentiating the identity of the two parks more clearly (option iii) would also help reduce confusion and provide a more accurate representation of the data.
Resizing the scale on the y-axis (option ii) may not be necessary in this case, as the existing scale is appropriate and accurately represents the data.
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If Wendy is 63 inches tall and her hand is 6. 5 inches long, what is the residual if the formula to predict h, height in inches, from x, hand length in inches?
If the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches
A residual is the difference between the predicted value of a variable (in this case, height) and the actual value of that variable. Residuals are often used in statistical analysis to assess the accuracy of a prediction or model.
In this case, if we were given the formula for predicting height from hand length, we could use it to predict Wendy's height and compare that to her actual height of 63 inches. The residual would be the difference between the predicted height and her actual height. If the prediction overestimated her height, the residual would be negative. If it underestimated her height, the residual would be positive.
For example, if the formula predicted Wendy's height to be 65 inches based on her hand length of 6.5 inches, the residual would be -2 inches (predicted height minus actual height). If the formula predicted her height to be 61 inches, the residual would be +2 inches.
Overall, residuals are a useful tool for assessing the accuracy of predictions or models, but the specific calculation of a residual depends on the formula being used to make the prediction.
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A real-world problem with a sample and a population is modeled by the proportion 66/100 = x/2,500
. Use the proportion to complete the sentences
The real-world problem is modeled by the proportion 66/100 = x/2,500, where 66 is the sample proportion and 2,500 represents the population size.
To find the value of x, which represents the number of individuals with a specific characteristic in the population, follow these steps:
1. Cross-multiply the terms in the proportion:
66 * 2,500 = 100 * x
2. Simplify the equation:
165,000 = 100x
3. Divide both sides by 100 to isolate x:
x = 1,650
Thus, 1,650 individuals in the population share the specific characteristic represented by the sample proportion. This proportion helps us understand and predict the prevalence of a certain characteristic or behavior within a larger population, based on the information gathered from a smaller sample.
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Part A
Alex has \$ 30,000$30,000 in his savings account that earns 10\%10% annually.
How much interest will he earn in one year?
Interest == \$$
Part B
If Alex spends 20\%20% of the interest received on buying furniture for his new house, what amount did he spent on furniture?
A) The amount of interest he will earn in a year is $3,000.
B) The amount he spent on furniture is $600.
Part A: To calculate the interest Alex will earn in one year, use the formula for simple interest:
Interest = Principal × Rate × Time.
In this case, Principal = $30,000, Rate = 10% (0.10), and Time = 1 year. So,
Interest = $30,000 × 0.10 × 1 = $3,000.
Part B: Alex spends 20% of the interest on furniture. To calculate this amount, multiply the interest by 20% (0.20): $3,000 × 0.20 = $600.
Therefore, in one year, Alex will earn $3,000 in interest. He will spend $600 on furniture for his new house.
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Consider the differential equation dy dx = 23 48 (A) Re-write the equation in terms of differentials: dy= dx LHS: RHS: (B) Now integrate each side of the equation: + C1 = = + C2 LHS: RHS: (C) Solve the equation for y, given that that y(0) = 4. Y=
The solution for the differential equation is y = (23/48) x + 4.
How to determined the ordinary differential equation?(A) Re-writing the differential equation in terms of differentials, we get:
dy = (23/48) dx
Here, dy and dx represent infinitesimal changes in the variables y and x, respectively.
(B) Integrating both sides of the equation with respect to their respective variables, we get:
∫dy = ∫(23/48)dx
On the left-hand side, the integral of dy is simply y (plus a constant of integration), while on the right-hand side, we can pull the constant factor (23/48) outside the integral:
y + C1 = (23/48) ∫dx
Integrating the right-hand side with respect to x, we get:
y + C1 = (23/48) x + C2
where C1 and C2 are constants of integration.
(C) To solve for y, we can isolate it on one side of the equation by subtracting C1 from both sides:
y = (23/48) x + (C2 - C1)
Next, we can use the initial condition y(0) = 4 to solve for the constant C2 - C1:
y(0) = (23/48) (0) + (C2 - C1) = C2 - C1
Since y(0) = 4, we have:
4 = C2 - C1
Therefore, C2 - C1 = 4, and we can substitute this back into the expression for y to get the final solution:
y = (23/48) x + 4
So the solution for the differential equation with initial condition y(0) = 4 is y = (23/48) x + 4.
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Mr. Smiths algebra class is inquiring about slopes of lines. The class was asked to graph the total cost, c, of buying h hotdog that cost 75 cent each. The class was asked to describe the slope between any two points on the graph. Which statement below is always a correct answer about the slope between any two points on this graph?
1)the same positive value
2)the same negative value
3) zero
4) a positive value, but the values vary
The slope of the graph is the same positive value that is 0.75.
Hence the correct option is (1).
We know that the equation of a straight line with slope 'm' and y intercept 'c' is given by,
y = mx + c
Here the model equation
c = 0.75h, where c is the total cost to buy hotdogs
h is the number of hotdogs bought
And 0.75 is the price of one hotdog
Now we can clearly say that c = 0.75h will make a straight line coordinate plane.
Now comparing the equation with slope intercept equation of straight line we get,
m = 0.75 and c = 0
So the slope of the line represented by model equation = 0.75 which is a positive number.
y intercept = 0.
We know that the slope of one particular straight line on cartesian plane is unique.
So, the slope of the graph is the same positive value.
Hence the correct option is (1).
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