the correlation coefficient (r) between the number of miles driven x and the number of gallons of gasoline used y is 0.925. what percent of the variation in the number of gallons of gasoline can be explained by differences in the number of miles driven?

The ordered pairs which correctly lists all of the x-intercepts of the graphed function include the following: C. (1, 0), (2, 0), and (–3, 0).

In Mathematics, the x-intercept can be defined as the point at which the graph of a particular function crosses the x-coordinate (x-axis) and the value of "y" is equal to zero (0).

By critically observing the given graphed function (see attachment), the x-intercept is given by the following ordered pairs:

This ultimately implies that, this function would cross the x-coordinate (x-axis) at points (1, 0), (2, 0), and (-3, 0).

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The wavelength of the given speed and frequency would be = 0.062m

Speed is defined as the distance a wave travels in a given amount of time, such that it is measured in meters per second.

The frequency of a wave is defined as the number of cycles that can be completed in a given time. This is measured in Hertz (Hz).

The wavelength is defined as the distance if a wave which can be measured in meters.

The formula for wave speed = v = fλ

Where v = Speed = 5.410x10⁴

f = 872khz = 872000hz

λ = ?

make λ the subject of formula;

λ = 5.410x10⁴/872000

λ = 54100/872000

λ = 0.062m

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(a) The impulse of the car of mass 1200 kg is 12000 Ns.

(b) The change in velocity of the car is 10 m/s.

Impulse is the product of force and time.

(a) To calculate the impulse of the force, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

(b) To calculate the change in velocity of the car, we use the formula below

Where:

From the question,

Given:

Substitute these values into equation 2

Hence the change in velocity of the car is 10 m/s.

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For given parametric curve the equations of both tangents at point where it crosses itself are: y = ± (√15x + 2√15)

The graph of parametric curve x = cos t + 2 cos 2t, y = sin t + 2 sin 2t (t ∈ [0, 2π]) is as shown below.

From the graph we can observe that, the curve crosses itself at (-2, 0).

Let x = -2

cos t + 2 cos 2t = -2,

and y = 0

sin t + 2 sin 2t = 0

sin t + 2 (2 sin(t) cos(t)) = 0

sin t + 4 sin t cos t = 0

sin t(1 + 4 cos t) = 0

sin(t) = 0   OR   1 + 4 cos(t) = 0

sin(t) = 0   OR   cos(t) = -1/4

From sin(t) = 0

⇒ t = 0 or t = π,

but for these values of t, cos t + 2 cos 2t = -2 is not true.

This means, the point (-2, 0) must corresponds to cos(t) = -1/4

[tex]\Rightarrow t=cos^{-1}(\frac{1}{4} )[/tex]

When cos(t) = -1/4,

sin(t) = [tex]\frac{\sqrt{15} }{4}[/tex]                ..............(by identity sin²(t) + cos²(t) = 1)

And

[tex]sin(2t) = 2sin(t)cos (t)\\sin(2t) = 2\times \frac{\sqrt{15} }{4} \times (\frac{-1}{4} )\\\\sin(2t) = -\frac{\sqrt{15} }{8}[/tex]

[tex]\\\\cos(2t)=2cos^2{t} - 1\\\\cos(2t)=2(\frac{-1}{4} )^2 - 1\\\\cos(2t)=\frac{1}{8}-1\\\\ cos(2t)=\frac{-7}{8}[/tex]

Now we find derivative of given parametric equations with respect to t.

for x = cos t + 2 cos 2t,

dx/dt = -sin(t) -4 sin(2t)

y = sin t + 2 sin 2t

dy/dt = cos(t) + 4 cos(2t)

Consider dy/dx

[tex]= \frac{\frac{dy}{dt} }{\frac{dx}{dt} }\\\\= \frac{cos(t) + 4 cos(2t) }{-sin(t) -4 sin(2t)}\\\\=\frac{\frac{-1}{4}+4~\frac{-7}{8} }{-\frac{\sqrt{15} }{4}-4~\frac{\sqrt{15} }{8} }\\\\=-\sqrt{15}[/tex]

This means, slope m = -√15

We have the point (−2,0)

Using equation of line y=mx+b the y-intercept would be,

0 = -√15 (−2) + b

b = -2√15

So, the equation for this tangent line :

y = -√15x - 2√15

And the equation of other tangent would be,

y = √15x + 2√15

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The quantity of the surullitos left over is 7.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that a chef makes 750 surulfitos and donates half of them to a soup kitchen. The rest are divided equally among 16 plates.

The leftover surulfitos are,

E =  750 / ( 2 x 16 )

E = 375 / 16

E = 23⁷/₁₆

The remainder is 7 so the leftover surullitos are 7.

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The location of B such that the ratio of AB to BC is given as follows:

B = 28.

The location of B in the context of this problem is obtained applying the proportion, which is the ratio between the locations.

The coordinates of each point in this problem are given as follows:

The length of each segment in this problem is given as follows:

The ratio of AB to BC is 3 to 1, hence the following proportional relationship is established:

AB/BC = 3.

(x - 4)/(36 - x) = 3.

Applying cross multiplication, we can solve to obtain the value of x, representing the location of B, as follows:

x - 4 = 108 - 3x

4x = 112

x = 112/4

x = 28.

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Answer:

You know that x = -9, and 3 or The first number is 3, the other -9

Step-by-step explanation:

First number: x

Second: x + 6

(x)^2 + (x + 6)^2 = 90

x^2 + x^2 + 6x + 6x + 36 = 90

Simplify more

2x^2 + 12x + 36 = 90

2x^2 + 12x - 54 = 0

2(x + 9)(x - 3) = 0

The area of triangle will be 140 square units.

The scale factor of a shape refers to the amount by which it is increased or shrunk. It is applied when a 2D shape, such as a circle, triangle, square, or rectangle, needs to be made larger.

K is the scaling factor for x in the equation y = Kx. This expression can also be visualised in terms of proportionality:

y ∝ x

K can therefore be regarded as a proportionality constant in this context.

Given the area of triangle = 35 sq. units

and scale factor is 2

The base times the height of a triangle equals its area. Triangle ABC's base and height each double when a scale factor of 2 with centre A is used, increasing the area by a factor of 4 as a result.

area of triangle = 1/2bh

where b = base and h = height

given 1/2bh = 35

when scale factor is 2

new base = 2b

new height = 2h

area will be 1/2(2b)(2h) = 4/2bh

so  1/2bh = 35

4/2bh = 35 x 4 = 140 square units.

Hence the area is 140 square units.

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312%, 312' 4', 13, 9, 4.2 x 10-2. This is a decending order.

What is meant by decending?

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Answer:

50%

Step-by-step explanation:

(new-old)/old × 100

(600 - 400)/400 × 100

(200)/400 × 100

0.5 × 100

50%

The charge for the t-shirt earlier than the cut price of 25 percent is $20.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given, the 25% discount, the price of a t-shirt was 15$.

Since, After 25% discount price of t-shirt = 15

Thus

Original price of t-shirt = 15 * 100/75

Original price of t-shirt = 20

Therefore, the price before the discount of 25 percent is $20.

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The solution to the system of equations for this problem is given as follows:

a. (1, -1, -4).

The system of equations for this problem is defined as follows:

The augmented matrix for the system is given as follows:

[tex]\left[\begin{array}{cccc}2&1&1&-3\\3&-5&3&-4\\5&-1&2&-2\end{array}\right][/tex]

To place in row-echelon form, first only the first row of the first column may have a diagonal different of zero, hence the transformation is given as follows:

Hence the matrix is given as follows:

[tex]\left[\begin{array}{cccc}2&1&1&-3\\0&-13&3&1\\0&-7&-1&11\end{array}\right][/tex]

Now the element at the third row and second column must be of zero, hence the transformation is of:

R3 = 13R3 - 7R2

Thus the matrix is of:

[tex]\left[\begin{array}{cccc}2&1&1&-3\\0&-13&3&1\\0&0&-34&136\end{array}\right][/tex]

Thus the solution for z is given as follows:

-34z = 136

34z = -136

z = -136/34

z = -4.

The solution for y is given as follows:

-13y + 3z = 1.

13y = 3z - 1

13y = -13

y = -1.

The solution for x is given as follows:

2x + y + z = -3.

2x = -3 - y - z

2x = -3 + 1 + 4

2x = 2

x = 1.

Meaning that option a is correct.

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The integral, ∫ [ (9ˣ - eˣ )/3ˣ ] dx evaluate value is 3ˣ/ ln(3) - eˣ/3ˣ(1 - ln 3). So, the correct answer is option (c).

Integration is defined as the process of finding the area of the region under the curve. This is done by drawing as many small rectangles covering up the area and summing up all their areas. If d/dx(F(x) = f(x), then ∫ f(x) dx = F(x) + C, where C is integration constant and indefinite integrals. We have an integral, say , I = ∫ [ (9ˣ - eˣ )/3ˣ ] dx

and we have to determine its value.

I = ∫ [ (9ˣ - eˣ )/3ˣ ] dx

=> I = ∫ [ (9ˣ/3ˣ - eˣ/3ˣ ] dx

=> I = ∫ [ (3²ˣ/3ˣ - eˣ/3ˣ ] dx

=> I = ∫ [ (3²ˣ/3ˣ - eˣ/3ˣ ] dx

=> I = ∫ [ (3ˣ - eˣ/3ˣ ] dx

=> I = ∫ 3ˣ dx - ∫ eˣ3⁻ˣ dx

=> I = 3ˣ/ ln(3) - eˣ/3ˣ(1 - ln 3) + C

( since, ∫aˣ dx = aˣ/ln a)

where C --> integration constant

So, the required value is 3ˣ/ln(3)-eˣ/3ˣ(1 - ln 3).

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Complete question:

Find integral ∫ [ (9ˣ - eˣ )/3ˣ ] dx .

a) 3ˣ - eˣ/(1 - ln 3) + C

b) 9ˣ/2ln3 - eˣ/(3ˣ ln 3) + C

c) 3ˣ/ln3 - eˣ/3ˣ(1 - ln 3) + C

d) 3ˣ/ln3 - eˣ/3ˣ + C

The square feet of the garden that could contain flowers, given the side lengths of the stepping stone and the garden, is C. 16 ft².

To find the portion of the garden that could contain flowers, first find the total area of the entire garden ;

= Length of garden x Width

= 4 x 6

= 24 ft ²

The stepping stones will not contain flowers so their area should be deducted from the area of the garden :

= 24 - ( 2 x ( 2 x 2 ))

= 24 - ( 2 x 4 )

= 24 - 8

= 16 ft²

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Answer:

y=mx+c

Step-by-step explanation:

For the distance of 5 miles covered in 4 hours, we can write the unit rate as {r] = 1.25 miles/hour.

The rate at which one value changes with respect to per unit change in the other value is called the unit rate. Mathematically -

{ r } = y/x                                                          .... Eq { 1 }

Given is the distance of 5 miles covered in 4 hours.

We can write the unit rate as -

{r} = 5/4

{r} = 1.25 miles/hour

Therefore, for the distance of 5 miles covered in 4 hours, we can write the unit rate as {r] = 1.25 miles/hour.

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Answer:

  (x/9)² +(y/3)² = 1 . . . P = (x, y)

Step-by-step explanation:

You want the equation of the locus of a point P that is 3 cm from the x-axis end of a 12 cm rod whose ends are on the x- and y-axes.

Let the x- and y-intercepts of the rod ends be represented by 'a' and 'b', respectively. The fixed length of the rod tells us ...

  a² +b² = 12²

according to the Pythagorean theorem.

The location of point P is 3/12 = 1/4 of the way from the x-intercept to the y-intercept. Its coordinates in terms of 'a' and 'b' are ...

  P = 3/4(a, 0) +1/4(0, b) = (3a/4, b/4)

If we define the point P as having coordinates (x, y), then we have ...

  3a/4 = x   ⇒   a = 4/3x

  b/4 = y   ⇒   b = 4y

Using these values in the above relation between 'a' and 'b', we have ...

  (4/3x)² +(4y)² = 12²

We can divide by 12² to get the following equation of the ellipse that is the locus of P.

  (x/9)² +(y/3)² = 1 . . . . . . useful domain/range: 0≤x≤9; 0≤y≤3.

[tex]~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{3(ab^2)^2}{6a^3b^{-3}}\implies \cfrac{3}{6}\cdot \cfrac{(ab^2)^2}{a^3b^{-3}}\implies \cfrac{1}{2}\cdot \cfrac{(a^2b^4)}{a^3b^{-3}}\implies \cfrac{1}{2}\cdot \cfrac{b^4 b^3}{a^{-2}a^3} \\\\\\ \cfrac{1}{2}\cdot \cfrac{b^{4+3}}{a^{-2+3}}\implies \cfrac{1}{2}\cdot\cfrac{b^7}{a^1}\implies \cfrac{b^7}{2a}[/tex]

Answer:

i think it's true am a 6 grader i really try

Step-by-step explanation:

We can find the discount by taking the product between the price of the couch and the decimal form of the percentage, the discount on the couch is equal to $258.75

We know that for a couch whose price is $1,725, we apply a discount of the 15%.

Then we just need to find the 15% of $1,725, to do so, we need tu multiply the number by the decimal form of that percentage.

To get the decimal form, you need to take the quotient between the percentage and 100%, you will get:

15%/100% = 0.15

Then the discount is:

0.15*$1,725 = $258.75

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Answer: 4 pie chart

Step-by-step explanation:

Each pie chart  [tex]\pi[/tex] wil have 4 parts to it  depending on  how the shape of the pie  is

When Zoey logged the grade level and instrument of every middle school student, she discovered that 28% of the eighth graders at the School of Rock play the guitar.

In mathematics, a ratio or number expressed as a fraction of 100 is called a percentage. Also occasionally used are the acronyms "pct.," "pct," and "pc." However, the percentage symbol "%" is widely used to represent it. The % quantity is dimensionally empty. Basically, percentages are fractions when the denominator is 100. A percent symbol (%) should be placed next to a number to indicate that it is a percentage. For instance, you score a 75% on a test if you properly answer 75 out of 100 questions (75/100). When calculating percentages, multiply the result by 100 after dividing the amount by the total. By multiplying (value/total) by 100%, the percentage is determined.

given

from the table

number of 8th grader = 64

8th grader playing guitar = 18

% of 8th grader playing grader = 18/64*100

= 0.28125*100

= 28.125%

When Zoey logged the grade level and instrument of every middle school student, she discovered that 28% of the eighth graders at the School of Rock play the guitar.

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Geometric series is convergent if the |r|<1 where r is the common ratio.

Let Sn=∑ni=0(−3/4)i then

Sn=(−3/4)n+1−1(−3/4)−1

Now take n→∞ then

Sn→0−1(−3/4)−1=4/7

because |−3/4|<1 and so (−3/4)n→0. Now note that your sum is

lim ∑i=1n+1(−3)i−14i=lim 14∑i=1n+1(−3)i−14i−1=1/4.lim Sn=1/7.

Geometric series: A geometric series is the result of adding together geometric sequences indefinitely. Depending on the sequence given to us, such infinite sums can either be finite or infinite. A series is considered to be convergent if the partial sums gravitate to a certain value, also known as a limit. In contrast, a divergent series is one whose partial sums do not reach a limit. Divergent series frequently reach, reach, or avoid a particular number.

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The number of coins of nickels and dimes will be 16 and 2, respectively.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

Jacob has $1 worth of nickels and dimes. He has a total of 18 nickels and dimes altogether.

Let 'x' be the number of nickels and 'y' be the number of dimes. Then the equations are given as,

x + y = 18                          ....1

0.05x + 0.10y = 1             ....2

From equations 1 and 2, then we have

0.05(18 - y) + 0.10y = 1

0.90 - 0.05y + 0.10y = 1

0.05y = 0.10

y = 2

Then the value of the variable 'x' is given as,

x + 2 = 18

x = 16

The number of coins of nickels and dimes will be 16 and 2, respectively.

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Jim will need to sail 4 miles across the lake from the marina to Kingston.

Miles is a unit of measurement used to measure the distance between two locations. It is equal to 1.609 kilometers and is commonly used in the United States and the United Kingdom. Miles are frequently used to measure the distance between two cities, the distance a person has traveled, or the length of a road or highway. It is also sometimes used to measure the distance of a flight.

Since he is driving 16 miles from the boat launch to the marina, he will have covered a total of 20 miles by the time he arrives at the marina. From there, it is 4 more miles to Kingston. Therefore, he will need to sail a total of 4 miles to reach his destination.

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The domain of (f/g)(x) in interval notation is [-3, 6].

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

From the graph,

f(x) has x values from x = -1 to x = 6

g(x) has x values from x = -3 to x = 5

The function f(x) is f(x) = mx + c

Find two coordinates from the graph of f(x).

= (-1, 3) and (6, 4)

m = (4 - 3) / (6 + 1) = 1/7

Now,

3 = (1/7) x -1 + c

3 = -1/7 + c
c = 3 + 1/7

c = 22/7

f(x) = (1/7)x + 22/7  _____(1)

The function g(x) is f(x) = mx + c

Find two coordinates from the graph of g(x).

= (-3, 2) and (5, 4)

m = (4 - 2) / (5 +3) = 2/8 = 1/4

Now,

2 = 1/4 x (-3) + c

2 = -3/4 + c
c = 2 + 3/4

c = 11/4

g(x) = (1/4)x + 11/4  _______(2)

Now,

From (1) and (2)

(f/g)(x)

= f(x)/g(x)

= ((1/7)x + 22/7) / ((1/4)x + 11/4)

The domain of f(x) and g(x) will include the domain of both f(x) and g(x).

So,

x = -1 to x = 6 and x = -3 to x = 5

This can be written as,

[-3, 6]

Thus,

The domain of (f/g)(x) is [-3, 6].

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∑ (-1)ⁿ xⁿ/n   is convergent for  x ∈ (−1,1].

Given the series ∑ (-1)ⁿ xⁿ/n

we can use the ratio test by evaluating:

n/(n+1) |x|

So we get two cases,

i) x = 1

∑ (-1)ⁿ xⁿ/n  = (-1)ⁿ/n which is the alternate harmonic series and is convergent.

ii) ∑ (-1)ⁿ xⁿ/n = ∑ (-1)ⁿ  (-1)ⁿ/n = ∑ 1/n which is the harmonic series and is divergent.

In conclusion the series is convergent for x ∈ ( -1, 1].

A power series' radius of convergence is equal to the diameter of the biggest disk at the series' convergence point. A non-negative real number or are both possible. When it is positive, the power series converges to the Taylor series of the analytical function absolutely and evenly on compact sets inside the open disk with radius equal to the radius of convergence.

The radius of convergence is the shortest or smallest of all the estimated distances from the center of the disk of convergence to each individual singularity of a function in the event of multiple singularities (singularities are those values of the argument for which the function is not defined).

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Answer:

------------------------------------

100 dozen add 20% is:

The value of x of the quadrilateral using the concept of similar quadrilaterals is; x = 1.5

Two quadrilaterals are referred to as similar quadrilaterals when the three corresponding angles are the same and as a result the fourth angles automatically become the same due to the fact that the interior angles of a quadrilateral sum up to 360 degrees and two adjacent sides have equal ratios.

Thus, we can say by the concept of similar quadrilaterals that;

AB/LM = BC/MN

Thus;

14/(x + 9) = 10/(x + 6)

14(x + 6) = 10(x + 9)

14x + 84 = 10x + 90

14x - 10x = 90 - 84

4x = 6

x = 1.5

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