We are given the parametric equations:
x = 6 cos θ
y = 6 sin θ
We know that the derivative of cos a = - sin a and the derivative of sin a = cos a, therefore taking the 1st and 2nd derivates of x and y:
d x = 6 (-sin θ) = - 6 sin θ
d^2 x = -6 (cos θ) = - 6 cos θ
d y = 6 (cos θ) = 6 cos θ
d^2 y = 6 (-sin θ) = - 6 sin θ
Therefore the values we are asked to find are:
dy / dx = 6 cos θ / - 6 sin θ = - cos θ / sin θ = - cot θ
d^2 y / d^2 x = - 6 sin θ / - 6 cos θ = sin θ / tan θ = tan θ
We can find the value of the slope at θ = π/4 by using the dy/dx:
dy/dx = slope = - cot θ
dy/dx = - cot (π/4) = - 1 / tan (π/4)
dy/dx = -1 = slope
We can find the concavity at θ = π/4 by using the d^2 y/d^2 x:
d^2 y / d^2 x = tan θ
d^2 y / d^2 x = tan (π/4)
d^2 y / d^2 x = 1
Since the value of the 2nd derivative is positive, hence the concavity is going up or the function is concaved upward.
Summary of Answers:
dy/dx = - cot θ
d^2 y/d^2 x = tan θ
slope = -1
concaved upward
subtract one- fifth from seven times w
An item is regularly priced at $89 . It is now priced at a discount of 65% off the regular price.
There are 36 girls and 40 boys at a summer camp. The campers will be divided into equal-sized groups. Each group will have the same number of girls and the same number of boys. What is the greatest number of groups that can be formed?
last year, Carmen opened an investment account with $7400. At the end of the year, the amount in the account had decrease buy 26.5%. How much is this decrease in dollars? How much money was in her account at the end of the year?
The decrease in Carmen's account was $1961, which is 26.5% of her original investment of $7400. Hence, by subtracting this decrease from the original investment, we find that Carmen had $5439 left at the end of the year.
Explanation:The decrease in Carmen's account over the year is 26.5% of the original investment of $7400. In mathematical terms, we can calculate this value by multiplying the original investment by the percentage decrease, using the formula: Decrease = Original Amount * Percentage Decrease.
In Carmen's case, this would be $7400 * 0.265 (converted from 26.5%). Upon calculating, we find that the decrease is $1961.
Now, to find how much money Carmen had at the end of the year, we subtract this decrease from the original amount invested. Again, using the formula: Final Amount = Original Amount - Decrease, we plug in the numbers to get $7400 - $1961, which equals $5439. This is how much Carmen had left in her account at the end of the year after the decrease.
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The sun produces 3.9 x 10^33 ergs of radiant energy per second. how many ergs of radiant energy does the sun produce in 3.25 x 10^3 seconds?
Answer:
1.2675 x 10^37
Step-by-step explanation:
you would have to first multiply 3.9 x 3.25 =12.675. It is not in scientific notation so you would have to change the decimal 1.2675 x 10^37
The ergs of radiant energy produced is: [tex]1.3* 10^{37}[/tex]
The rate of energy is given as:
[tex]Rate = 3.9 * 10^{33}\[/tex] radiant energy per second
The time is given as:
[tex]Time = 3.25 * 10^3[/tex] seconds
The amount of radiant energy produced in this time is:
[tex]Energy = Rate * Time[/tex]
So, we have:
[tex]Energy = 3.9 * 10^{33} * 3.25 * 10^3[/tex]
Multiply
[tex]Energy = 12.675* 10^{36}[/tex]
Rewrite as:
[tex]Energy = 1.2675* 10^{37}[/tex]
Approximate
[tex]Energy = 1.3* 10^{37}[/tex]
Hence, the ergs of radiant energy produced is: [tex]1.3* 10^{37}[/tex]
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How do you rename 5400
What is an open map?
Find a vector function, r(t), that represents the curve of intersection of the two surfaces. the paraboloid z = 7x2 + y2 and the parabolic cylinder y = 3x2
Final answer:
The vector function r(t) representing the curve of intersection of the surfaces z = 7x² + y² and y = 3x² is r(t) = it + 3t²j + (7t² + 9t´)k.
Explanation:
Finding a Vector Function for a Curve of Intersection
To find the vector function r(t) that represents the curve of intersection of the surfaces z = 7x² + y² and y = 3x², we can parameterize the variables using a suitable parameter, commonly denoted as t. We can set x = t, which imminently provides y as y = 3t² by substituting t into the equation of the parabolic cylinder. Then, substituting both x and y into the equation of the paraboloid gives us z = 7t² + (3t²)² = 7t² + 9t´. Therefore, the vector function is:
r(t) = it + j3t² + k(7t² + 9t´).
This function r(t) represents all points (x, y, z) on the curve where the two surfaces intersect. Since the curve lies on both surfaces simultaneously, z must be equal for both equations when x and y from the curve are substituted into them.
To test μ for an x distribution that is mound-shaped using sample size n ≥ 30, how do you decide whether to use the normal or student's t distribution?
someone help me. Not good at math
perimeter = add all 3 sides 13+13+10 = 36 cm
for area you need to find the height of the triangle, which is the red dotted line
to do that use the Pythagorean Theorem
a^2 + B^2= c^2
a^2 = height
a^2 +5^2 = 13^2
a^2 +25 = 169
a^2 = 169-25 = 144
a=square root(144) = 12
so the height is 12
area = 1/2 h x b = 1/2 x 12 x 10 = 1/2x 120 = 60
area = 60 cm^2
cost is $4 per square cm
there is 60 square cm in the triangle so multiply 60 x 4 = $240
What is the derivative of this function?
Find the missing coordinate if the ordered pair is to be a solution to the equation 3x+y=9:(?,-9)
The vertex form of the equation of a parabola is y = (x - 4)2 + 22. What is the standard form of the equation?
The equation in standard form is y = x^2 - 8x + 38
How to write the equation in standard form?The vertex form of the equation is given as':
y = (x - 4)^2 + 22
Expand the equation
y = x^2 - 8x + 16 + 22
Evaluate the sum
y = x^2 - 8x + 38
Hence, the equation in standard form is y = x^2 - 8x + 38
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In mathematics, the distance between one point (a) and another point (b), each with coordinates (x,y), can be computed by taking the differences of their x coordinates and their y coordinates and then squaring those differences. the squares are added and the square root of the resulting sum is taken and... voila! the distance. given two variables, p1 and p2 that are of type point -- a structured type with two fields, x and y, both of type double-- write an expression whose value is the distance between the two points represented by p1 and p
The distance between two points can be found using the formula sqrt((x2 - x1)^2 + (y2 - y1)^2).
Explanation:The distance between two points can be found using the formula sqrt((x2 - x1)^2 + (y2 - y1)^2). To find the distance between the two points represented by p1 and p2, we substitute the x and y coordinates of p1 and p2 into the formula:
distance = sqrt((p2.x - p1.x)^2 + (p2.y - p1.y)^2).
What is the ratio of the two values and what new value do they produce? 4m in 2 min
Answer:
2:1
Step-by-step explanation:
The ratio is 2:1 and the new value can also be 8/4= 2/1
1. Derive the quadratic formula from the standard form (ax2 + bx + c = 0) of a quadratic equation by following the steps below.
2. Divide all terms in the equation by a.
3. Subtract the constant (the term without an x) from both sides.
4. Add a constant (in terms of a and b) that will complete the square.
5. Take the square root of both sides of the equation.
6. Solve for x.
We are given that:
a x^2 + b x + c = 0
Divide all terms with c/ a:
x^2 + (b / a) x + (c / a) = 0
Subtract c from both sides:
x^2 + (b / a) x + (c / a) – c / a = 0 – c / a
x^2 + (b / a) x = - c / a
Add a constant k to complete the square:
where k = ((b / a) / 2)^2 = (b /2a)^2
x^2 + (b / a) x + k = - c / a + k
x^2 + (b / a) x + (b/2a)^2 = - c / a + (b /2a)^2
So the perfect square trinomial is:
(x + b/2a)^2 = - c / a + (b /2a)^2
Taking the square root of both sides:
x + b/2a = sqrt [- c / a + (b /2a)^2]
x = sqrt [(-c/a) + (b /2a)^2] – (b/2a)
Answer:
My equation is 3x^2 + 12x + 24 = 0. When I divide all of the terms by 3, the equation becomes x^2 + 4x + 8 = 0. When I subtract both sides by the constant, 8, the equation is x^2 + 4x = -8. When I add a constant to create the perfect square, the equation now becomes x^2 + 12x + 36 = 28. Finally, the equation simplified equals (x + 6)= the square root of 28.
Step-by-step explanation:
805 tens in standard form
A balloon is released and flies straight up for 120 feet, and then the wind blows it east for 280 feet. Find the distance between the balloon and the site where it was released. Decimal rounded one place.
Find a formula for the inverse of the function, f(x)= 8-3/x^2, x>0. Explain if you can.
Timothy explains to his brother that the model y=4mt has a constant of variation of 4 and is an example of _____ variation.
Answer:
Joint variation.
Step-by-step explanation:
There are four types of variation,
Direct variation : When a variable varies directly as another variable,
Example : y varies directly with x ⇒ y = kx,
Where, k is the constant of variation,
Inverse variation : When a variable varies inversely as another variable,
Example : y varies inversely with x ⇒ [tex]y=k\frac{1}{x}[/tex]
Joint variation : When a variable varies directly with two or more than two variables,
Example : y varies directly with both x and z ⇒ y = kxz
Combined variation : When a variable varies both directly and indirectly as another variables,
Example : y varies directly with x and inversely with z ⇒ [tex]y=k\frac{x}{z}[/tex]
Now, here the given expression,
y = 4mt,
Where, y, m and t are variables and 4 is the constant of variation,
⇒ y varies directly as both m and t,
⇒ y = 4mt is an example of joint variation.
Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p.
n=128,
p=0.27
The mean is ______
. (Round to the nearest tenth as needed.)
The variance,
σ^2 is ______
(Round to the nearest tenth as needed.)
The standard deviation, ______
A binomial distribution is an experiment where there are two outcomes; Success and failure. In here, you are given n = 128 and p = 0.27. The formula for the variance of a binomial distribution is n*p(1 – p) or n*p*q*. They are equivalent because q = 1- p. We will use the first equation.
Variance = n*p*(1 – p)
Variance = 128*0.27*(1 – 0.27)
Variance = 25.23
To get the standard deviation, find the square root of the variance.
Standard deviation = sqrt (Variance)
Standard deviation = sqrt (25.23)
Standard deviation = 5.02
The slope of a vertical line is zero. true or false
The solution is False
The slope of a vertical line is given by the equation Slope m = undefined
What is an Equation of a line?
The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the line be represented as A
Now , the line A is a vertical line
So , the equation of line will be represented as y = mx + c
where m is the slope of the line
And , Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
The slope of a vertical line is infinite or undefined as it has no y-intercept
Because , the change is x is 0
So , the denominator of the fraction is 0 and division by 0 is undefined
Slope m = ( y₂ - y₁ ) / 0
Hence , the slope m is undefined
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Knowing that this number, the Golden Ratio, is present not just in mathematics, but may also be present within your own brain and body (at the atomic or subatomic level), what do you think it means? Is this number evidence of a grand design, a massive freak coincidence or something else?
The Golden Ratio is a mathematical relationship that exists in art, shapes, nature and the human body. The golden ratio can be present in your body, from the length of your arms and legs when compared to your torso. Fingers is another example because the length of our fingers, each section from the tip of the base to the wrist is larger than the preceding.
The measurement of the human navel to the floor and to the top of the head to the navel is also the Golden ratio. Plastic surgeons and dental surgeons use it to reconstruct the human face. It also appears in everything around us like in the nature and science. It appears on in flower petals because it is believed that each petal is placed to so that each petal gets the best exposure to sunlight. Dolphins, starfish, sea urchins and honeybees also exhibit the proportion like humans. DNA molecules measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral, these two number are successive numbers. I think that the Golden Ratio is just a freak coincidence that happened.
whats the equation for
4n + 6 - 2n= 2 (n+3)
At a concert there needs to be at least 1 member of staff working for every 150 members of the audience. If there are 20 members of staff, how many people are in the audience?
Henry recorded the number of miles he biked each day for a week. His miles were 25, 40, 35, 25, 40, 60, and 75. Enter the data into the statistics calculator. What is the standard deviation of the miles Henry biked to the nearest tenth?
The standard deviation of the miles Henry biked to the nearest tenth is 17.1. The standard deviation is calculated following the steps below.
Calculation of standard deviationFirst calculate the mean of the numbers that where given which is = 25 + 40 + 35 + 25 + 40 + 60+ 75/7
= 300/7
= 42.9
Then for each number, subtract the Mean and square the result.
(25 - 42.9)² = −17.9² = 320.41
(40-42.9)² = -2.9²= 8.42
(35-42.9)² = -7.9² = 62.41
(25-42.9)² = −17.9² = 320.41
(40-42.9)² = -2.9²= 8.42
(60-42.9)² = 17.1²= 292.41
(75-42.9)² = 32.1²= 1030.41
Then work out the mean of those squared differences. That is,
320.41+ 8.42+62.41+320.41+8.42+292.41+1030.41/7
= 2042.89/7
= 291.8
Take the square root of 291.8 which is,
√291.8 = 17.08215443
= 17.1 to the nearest tenth.
Therefore, the standard deviation of the miles Henry biked to the nearest tenth is = 17.1
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Answer:
17.1
Step-by-step explanation:
got it right
Nate rides his bike to school every day. From his home he rides 2.5 miles north, and then six miles west. If a straight line is drawn from Nate's house to the school, find the distance between these two locations.
use Pythagorean Theorem
2.5^2 + 6^2 = x^2
6.25 +36 = 42.25
x= sqrt(42.25) = 6.5 miles
in 2000 you weighed 175 pounds. in 2001 you weighed 62 pounds, and in 2002 you weighed 154 pounds. how many pounds did you lose from 2000 to 2002.
Stella is renting a car for a business trip. The agency charges a flat rate of $60 plus $0.50 per mile every mile she drives over 40 miles. Stella's company will pay up to $270, excluding the cost of gasoline , for the use of the rental car. What is the number of miles Stella can drive the rental car without spending more then$270
270 =60 + 0.50(x-40)
270 =60 + 0.50x -20
210 =0.50x -20
230=0.50x
x=230/0.50 = 460 miles
check:
460-40 = 420
420*0.50=210 +60=270
she can drive 460 miles total
A moving company charges 0.60 per pound for a move from New York to Florida. A family estimates that their belongings weigh about 4 tons. About how much would it cost the family to move from New York to Florida?