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1) y= 2 x- 4 find parallel line through this point (-4,7)

2) y= -6x find parallel line through this point ( 4,9)

3) y= -½ x +3 find the perpendicular through this point ( 0,8)

4) y= -⅓ x find the perpendicular through this point (-2,1)

Vector equation of the line tangent to the graph of r(t) at the point p0 on the curve r(t) = (3t - 1) i + 13t j + 4 k.

To find a vector equation of the line tangent to the graph of r(t) at the point P0 on the curve r(t) = (3t - 1) i + 13t j + 16 k, where P0 is given as (-1,4), we can use the following steps:

Step 1: Find the derivative of r(t) with respect to t:

r'(t) = 3 i + 13 j

Step 2: Evaluate the derivative at the point P0:

r'(-1) = 3 i + 13 j

Step 3: Use the point P0 and the vector r'(-1) to form the vector equation of the tangent line:

r(t) = P0 + r'(-1) t

where t is a scalar parameter.

Plugging in the values, we get:

r(t) = (-1)i + 4j + (3i + 13j)t

Simplifying, we get:

r(t) = (3t - 1) i + 13t j + 4 k

Therefore, the vector equation of the line tangent to the graph of r(t) at the point P0 on the curve

r(t) = (3t - 1) i + 13t j + 16 k  is

r(t) = (3t - 1) i + 13t j + 4 k.

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Answer: B. statistic

Step-by-step explanation: A characteristic, usually a numerical value, which describes a sample, is called a statistic.

The probability that at least one of the lights is defective is 0.2661 .

in the question ,

it is given that

the total number of bulbs in the pack of new light bulbs is (n) = 5

percent of bulb that did not work is  = 6%

probability that lights did not work is (p) = 0.06

probability that lights work is (q) = 1 - 0.06 = 0.94

let x be the number of defective lights

So ,the probability that at least one of your lights is​ defective

is P(x ≥ 1) = 1 - P(x<1)

= 1 - P(x = 0)

By Binomial Probability

= 1 - ⁿCₓ*(p)ˣ*(q)ⁿ⁻ˣ

= 1 - ⁵C₀*(0.06)⁰*(0.94)⁵⁻⁰

= 1 - 1*1*(0.94)⁵         ...because ⁵C₀ = 1

= 1 - (0.94)⁵

= 1 - 0.7339

= 0.2661

Therefore , The probability that at least one of the lights is defective is 0.2661 .

The given question is incomplete , the complete question is

You purchased a​ five-pack of new light bulbs that were recalled because ​6% of the lights did not work. What is the probability that at least one of your lights is​ defective ?

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Answer:

r = \(\frac{29x}{2\pi }\)

x = \(\frac{2\pi r}{29}\)

Step-by-step explanation:

C = 2\(\pi\)r

29x = 2\(\pi\)r  --------- (i)

divide both sides of (i) by 2\(\pi\)

r =  \(\frac{29x}{2\pi }\)

but when you divide both sides of (i) by 29, you get

x = \(\frac{2\pi r}{29}\)

Answer:

Step-by-step explanation:

clay will have

-80+96=16 dollars

the change in balance is +16

Answer:

No

Step-by-step explanation:

x=(b-5)/c≠(5-b)/c

supposedly

b=6

c=1

x=6-5/1=1

x=5-6/1= -1

1≠ -1

Answer:

No

Step-by-step explanation:

x = b - 5 / L equal to - (5 - b) / L

The correct answer is False. The non-negativity of the double integral does not guarantee that the function itself is non-negative for all points in the region.

The inequality ∬_Df(X,Y)DA ≥ 0 represents the condition that the double integral of the function f(X, Y) over the region D is greater than or equal to zero. This means that the net "signed" area under the surface defined by the function f(X, Y) over the region D is either positive or zero.

However, this condition does not imply that the function f(X, Y) itself is non-negative (greater than or equal to zero) for all points (X, Y) within the region D. The function f(X, Y) may take both positive and negative values at different points in D, as long as the overall integral over the region is non-negative.

To illustrate this, consider a simple example where the region D is a circular disk centered at the origin. Let's assume that the function f(X, Y) is defined as f(X, Y) = X - Y. In this case, f(X, Y) can take negative values within the disk, but the double integral ∬_Df(X,Y)DA could still be non-negative if the positive and negative contributions balance out.

So, while the non-negativity of the double integral guarantees that the net signed area is non-negative over the entire region, it does not guarantee that the function itself is non-negative for all points in the region. Therefore, the statement "F(X,Y) ≥ 0 for all (X,Y) in D" is not necessarily true.

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Answer:

\( \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2( {x}^{2} - {y}^{2}) }{ {x}^{2} - {y}^{2} } = \boxed{ \displaystyle \frac{4y ^2}{(x - y)(x + y)} }\)

Step-by-step explanation:

we want to simplify the following

\( \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2( {x}^{2} - {y}^{2}) }{ {x}^{2} - {y}^{2} }\)

notice that we can reduce the fraction thus do so:

\( \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - \frac{2 \cancel{( {x}^{2} - {y}^{2}) }}{ \cancel{{x}^{2} - {y}^{2} }}\)

\( \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - 2 \)

in order to simplify the addition of the algebraic fraction the first step is to figure out the LCM of the denominator and that is (x-y)(x+y) now divide the LCM by the denominator of very fraction and multiply the result by the numerator which yields:

\( \rm \displaystyle \frac{x + y}{x - y} + \frac{x - y}{x + y} - 2 \\ \\ \displaystyle \frac{(x + y)^2 + (x - y)^2 - 2(x + y)(x - y)}{(x - y)(x + y)} \)

factor using (a-b)²=a²+b²-2ab

\( \rm \displaystyle \frac{(x + y-(x - y) )^2}{(x - y)(x + y)} \)

remove parentheses

\( \rm \displaystyle \frac{(x + y-x + y) )^2}{(x - y)(x + y)} \)

simplify:

\( \rm \displaystyle \frac{4y ^2}{(x - y)(x + y)} \)

9514 1404 393

Answer:

  4y²/(x² -y²)

Step-by-step explanation:

The expression simplifies as follows:

  \(\dfrac{x+y}{x-y}+\dfrac{x-y}{x+y}-\dfrac{2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{(x+y)(x+y)+(x-y)(x-y)-2(x^2-y^2)}{(x-y)(x+y)}\\\\=\dfrac{(x+y)^2+(x-y)^2-2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{(x^2+2xy+y^2)+(x^2-2xy+y^2)-2(x^2-y^2)}{x^2-y^2}\\\\=\dfrac{2(x^2+y^2-(x^2-y^2))}{x^2-y^2}=\boxed{\dfrac{4y^2}{x^2-y^2}}\)

Answer:

Step-by-step explanation:

A store offers discounts on all its products at the same percentage. Camila bought a beach chair that cost $ 26,970, but paid $ 21,576. If he also paid $ 15,992 for a sunshade. What was the original price of this one?

Step 1

Find the percentage discount

% discount = Discount amount/ Original amount × 100

= $26,970 - $ 21,576/$26,970 × 100

= 0.2 × 100

= 20%

Answer:

In D=(−∞,−3]∪[3,+∞), the range of x→x2−9 is [0,+∞).

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

To find a perpendicular line, you must compare the slopes. With the equation y=2x-5, we're looking for the inverse/opposite slope of -1/2.

Option A still has a slope of 2. Option B, when everything is in the format of mx+b = y format is -x-4=-2y. which gives us 1/2 but not -1/2. C when everything is on the right side gives us -2y=x-6. Divide both sides by -2 to get the y by itself and you're left with y = -1/2x+3. Now we found the equation that gives us the -1/2 slope we were looking for.

Answer:

the city at a lower altitude would have a colder climate

The direction and by how many units should f(x) be shifted to match g(x) is Left by 6 units. Option C

To determine the direction and magnitude of the shift needed for f(x) to match g(x), we can compare the corresponding values of f(x) and g(x) in the table.

For x = -2, f(x) = 4 and g(x) = 64.

For x = -1, f(x) = 1 and g(x) = 49.

For x = 0, f(x) = 0 and g(x) = 36.

For x = 1, f(x) = 2 and g(x) = 25.

For x = 2, f(x) = 4 and g(x) = 16.

By comparing the values, we can observe that the graph of f(x) is shifted to the right compared to g(x). To match the graph of g(x), we need to shift f(x) to the left.

The magnitude of the shift can be determined by the difference in x-values between the corresponding points of f(x) and g(x).

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Answer:

54%

Step-by-step explanation:

Definition

Theoretical Probability is the ratio of the number of favorable outcomes to the total possible outcomes of an event.

Expressing probability as a formula:

Information about a standard deck of cards:

A standard deck of cards has four suits: hearts, clubs, spades, diamonds. Each suit has thirteen cards: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. Thus the entire deck has 52 cards total.

(a) Picture card (J, Q, and K)

In a standard deck, there are 4 Jacks, Queens, and Kings. Hence, the probability of J, Q, and K are:

Probability as a fraction:

In decimal:

Percent:

(b) 9 or 10

There are 4 cards labeled 9 and 10 each.

Probability as a fraction:

In decimal:

In percent:

(c) 2 of heart

In a standard deck of cards, there are only one 2 of hearts.

Probability as a fraction:

In decimal:

In percent:

Answer:

Step-by-step explanation:

this is independent  

20/50= 2/5= 40% of getting pink, probability of getting 2= 20%

13/50=0.26= 26% of getting blue, probability of getting 2= 13%

17/50=0.34=34% of getting purple, probability of getting 2 = 17%

70%

Answer:

A

Step-by-step explanation:

the order of letter should resemble the same shape

Answer:

90 days

Step-by-step explanation:

Let the rate per day for the 30 days period be x

Given that the rate drops to 1/3 of the rate used in 30 days period, it means that the new rate

= 1/3 x

If at a rate of x, the full tank runs for 30 days

at a rate of 1/3 x, the full tank would run for

= x/ (1/3 x) * 30 days

= 3 * 30 days

= 90 days

At the new rate, the full tank of oil will last for 90 days

The likelihood that the station will need 13 or more calls to determine a winner is 0.5 because each call is independent.

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The probability that the mean of our sample is greater than $70 is 0.62 or 62%.

The probability of an event occurring is obtained by dividing the number of expected outcomes by the total number of events. In the question provided, we have a normal distribution, which means that we need the z score and this is gotten with the formula:

z = X - μ/σ

= 70 - 65/2

= 2.5

Now we find the P value for 2.5 and this is 0.9938

1 - 0.9938

= 0.0062

= 62%

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Answer: 12 cm^2

Step-by-step explanation:

Find area of small square = 2x2 = 4cm^2

Find Area of Big Square = 4x4 = 16cm^2

Area of Shaded Region = Area of Big Square - Area of small square = 16 - 4 = 12cm^2

the volume in the first octant bounded by\(y^2=4−x\) and y=2z is pi/36 sqrt(3).

(1) To find the volume in the first octant bounded by the surfaces \(y^2 = 4 - x\) and y = 2z, we can set up a triple integral in cylindrical coordinates.

First, we need to determine the bounds for our variables. Since we are working in the first octant, we know that 0 <= z, 0 <= theta <= pi/2, and 0 <= r.

Next, we need to find the equation for the upper and lower bounds of z in terms of r and theta. We can start with the equation \(y^2 = 4 - x\) and substitute y = 2z to get:

\((2z)^2 = 4 - x\)

\(4z^2 = 4 - x\)

\(x = 4 - 4z^2\)

We can then use this equation along with the equation z = y/2 to get the bounds for z:

\(0 < = z < = (4 - x)^(1/2)/2 = (4 - 4z^2)^(1/2)/2\)

Squaring both sides, we get:

\(0 < = z^2 < = (1 - z^2)/2\)

\(0 < = 2z^2 < = 1 - z^2\)

\(z^2 < = 1/3\)

So the bounds for z are:

\(0 < = z < = (1/3)^(1/2)\)

Finally, we can set up the triple integral in cylindrical coordinates:

V = ∫∫∫ r dz dtheta dr

with bounds:

0 <= r

0 <= theta <= pi/2

\(0 < = z < = (1/3)^(1/2)\)

and integrand:

r

So the volume in the first octant bounded by y^2=4−x and y=2z is:

V = ∫∫∫ r dz dtheta dr

= ∫ from 0 to\((1/3)^(1/2) ∫ from 0 to pi/2 ∫ from 0 to r r dz dtheta dr\)

= ∫ from 0 to\((1/3)^(1/2) ∫ from 0 to pi/2 r^2/2 dtheta dr\)

= ∫ from 0 to\((1/3)^(1/2) r^2 pi/4 dr\)

\(= pi/12 (1/3)^(3/2)\)

= pi/36 sqrt(3)

Therefore, the volume in the first octant bounded by\(y^2=4−x\) and y=2z is pi/36 sqrt(3).

(2) To find the volume bounded by z = x^2 + y^2 and z = 4, we can use a triple integral in cylindrical coordinates.

First, we need to determine the bounds for our variables. Since we are working in the region where z is bounded by \(z = x^2 + y^2\) and z = 4, we know that 0 <= z <= 4.

Next, we can rewrite the equation \(z = x^2 + y^2\) in cylindrical coordinates as \(z = r^2.\)

So the bounds for r and theta are:

0 <= r <= 2

0 <= theta <= 2pi

And the bounds for z are:

\(r^2 < = z < = 4\)

Finally, we can set up the triple integral in cylindrical coordinates:

V = ∫∫∫ r dz dtheta dr

with bounds:

0 <= r <= 2

0 <= theta <= 2pi

\(r^2 < = z < = 4\)

and integrand: 1

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In a rectangle where the length is 3 times the width, and the perimeter is 3 times the width. The perimeter is 96 cm, the length is 36 cm and the width is 12 cm.

Given,

Length of a rectangle is 3 times the width.

Perimeter is 3 times the width.

Perimeter is 96 cm.

Let w be the width of the rectangle.

The length of the rectangle is 3 times the width, so the length l is 3w.

The perimeter of the rectangle is equal to the sum of all its sides, so:

2w + 2(3w) = 96

2w + 6w = 96

8w = 96

w = 12

Therefore, the length is 36 cm and the width is 12 cm.

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To eliminate the roots in 1.1 and 1.2, we can use trigonometric substitution. In 1.1, we can substitute x = 4 sin(theta) to eliminate the root of 4. In 1.2, we can substitute z = 7 sin(theta) to eliminate the root of 7.

1.1. V64+2 + 1 We can substitute x = 4 sin(theta) to eliminate the root of 4. This gives us:

V64+2 + 1 = V(16 sin^2(theta) + 2 + 1) = V16 sin^2(theta) + V3 = 4 sin(theta) V3 1.2. V4z2 – 49

We can substitute z = 7 sin(theta) to eliminate the root of 7. This gives us:

V4z2 – 49 = V4(7 sin^2(theta)) – 49 = V28 sin^2(theta) – 49 = 7 sin(theta) V4 – 7 = 7 sin(theta) (2 – 1) = 7 sin(theta)

Here is a more detailed explanation of the substitution:

In 1.1, we know that the root of 4 is 2. We can substitute x = 4 sin(theta) to eliminate this root. This is because sin(theta) can take on any value between -1 and 1, including 2.

When we substitute x = 4 sin(theta), the expression becomes V64+2 + 1 = V(16 sin^2(theta) + 2 + 1) = V16 sin^2(theta) + V3 = 4 sin(theta) V3

In 1.2, we know that the root of 7 is 7/4. We can substitute z = 7 sin(theta) to eliminate this root. This is because sin(theta) can take on any value between -1 and 1, including 7/4.

When we substitute z = 7 sin(theta), the expression becomes: V4z2 – 49 = V4(7 sin^2(theta)) – 49 = V28 sin^2(theta) – 49 = 7 sin(theta) V4 – 7 = 7 sin(theta)

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Answer = 2 minutes and 15 seconds

1 minute = 60 seconds

60 seconds times 3= 180 seconds

25% of 180 seconds = 45 seconds

180-45=135 seconds

convert into minutes

1 minute = 60 seconds

2 minutes = 120 seconds

135-120=15 seconds

2 minutes and 15 seconds

Answer:

2 mins 15 secs

Step-by-step explanation:

If the original version of the song is 100% in length, then the remix will be 75% of the length of the original, as:

100% - 25% = 75%

Convert 75% into a decimal:

⇒ 75% = 75/100 = 0.75

75% of 3 minutes

= 0.75 × 3

= 2.25 minutes

As there are 60 seconds in a minute,

= 0.25 of a minute = 0.25 × 60 = 15 seconds

Therefore, the length of the remix is 2 mins 15 secs

2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).

To prove that the expression 2n^3 + 3n + 10 is in the set Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3), we need to show that the expression can be written in the form a(n^3) for some constant "a".

Let's start by factoring out the common factor of n^3 from each term:

2n^3 + 3n + 10 = n^3(2 + 3/n^2 + 10/n^3)

Now, let's rewrite the expression as a single term multiplied by n^3:

2n^3 + 3n + 10 = (2 + 3/n^2 + 10/n^3)n^3

Simplifying the expression inside the parentheses:

= (2n^3 + 3n^2 + 10n^3)/n^3

= (12n^3 + 3n^2)/n^3

= 12 + 3/n

ow, we can see that the expression can be written in the form a(n^3), where a = 12 and n^3 = 3/n.

Therefore, we have shown that 2n^3 + 3n + 10 can be written as a polynomial of the form Q(n^3), where Q(n^3) represents the set of polynomials of the form a(n^3).

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