Given f (x) = 3x + 15, find a when f (x) = 51.

Answer:

Look at pic 2 for the measurements. I rounded to the thousandth, because some required rounding to the hunderedth and others required to the nearest mile and others were unspecified.

Step-by-step explanation:

We should first draw a diagram. Since you said a diagram is shown, I will assume that the diagram looks something like pic 1. I do not know which variables correspond to which lengths, so I used random variables, but the diagram I drew should help you figure out which variables I used correspond to your actual problem.

If so, we can use our trigonometric functions and Pythagorean theorem to solve for all the lengths

We must solve for x and y in the diagram, and we can do so by using sine and cosine. We know the orange angle of the triangle whose legs are x and y is 36 degrees. We also know the hypotenuse of that triangle is of length 20 nm. Thus, [tex]sin(36) = \frac{x}{20}[/tex] and [tex]cos(36) = \frac{y}{20}[/tex].

We can multiply both sides of both of these functions by 20 to get: [tex]x = sin(36) * 20[/tex] and [tex]y = cos(36) * 20[/tex]. These are approximately x = 11.756 nautical miles and y = 16.180 nautical miles

We can solve for the length of a by adding 40 to y and getting a = 16.180 + 40 = 56.180 nautical miles

We can solve for the length of z by using the Pythagorean theorem and the leg lengths of a and x. We can do [tex]\sqrt{(40 + 20cos(36))^2 + (20sin(36))^2}[/tex] and get z = [tex]\sqrt{3294.427191}[/tex] = 57.397 nautical miles

Thus, we get the measurements in pic 2 as the distances you are asked to solve for.

The lengths are 56.180 nautical miles and 57.397 nautical miles

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”. The sides of this triangle have been named Perpendicular, Base, and Hypotenuse.

Given that, A ship sails 40 nautical miles (nm) due west from point A and then changes course and sails 20 nautical miles in a direction that is 54° west of due north.

We can use our trigonometric functions and Pythagorean theorem to solve for all the lengths

We must solve for x and y in the diagram, and we can do so by using sine and cosine. Furthermore, we also know the hypotenuse of that triangle is of length 20 nm. Thus,  and.

We can multiply both sides of both of these functions by 20 to get:  and. These are approximately x = 11.756 nautical miles and y = 16.180 nautical miles.

We can solve for the length of a by adding 40 to y and getting a = 16.180 + 40 = 56.180 nautical miles

We can solve for the length of z by using the Pythagorean theorem and the leg lengths of a and x. Likewise, we can do √40+(20cos(36°)²)+(20sin36°)²) and get z = 57.397 nautical miles.

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Answer:

Red=X-Int: (3,0) Y-int: (0,-5)

Step-by-step explanation:

To find the y and x intercepts you have to plug in the 0 for the variable. You'd get 5(0)-3y=15 you get this by plugging in the 0 for x and then you solve. 5 x 0 is 0 so you'd get -3y=15 divide -3 on both sides and get y=--5 meaning that is the y-intercept. You'd do the same when you look for the x-int. Just that you'd plug in a 0 for y instead of x. Which is 5x-3(0)=15. -3 x 0 is 0. So you'd have 5x=15 divide both sides by 5 and you'd get x=3 meaning that is the x intercept.

(Hopefully you were able to understand how I got the x and y intercepts for this equation)

The points on the ellipse that lie closest to the origin are  (1, 0, 0), (0, 1, 0)

Let (x, y, z) be a point on an ellipse in which plane x + y + z = 1 cuts the cylinder [tex]$x^2+y^2=1$[/tex]

Let f(x, y, z) = [tex]$x^2+y^2+z^2$[/tex]

[tex]& g_1(x, y, z)=x^2+y^2-1=0 \\[/tex]

[tex]& g_2(x, y, z)=x+y+z-1=0[/tex]

From Lagrange Multiplier Method.

There are multiple ways to define the method of Lagrange multipliers.

Suppose f and g are two functions such that they both have continuous partial derivatives.

Let (x0, y0, z0) ∈ S := {(x, y, z) : g(x, y, z) = 0} and ∇g(x0, y0, z0) ≠ 0.

The method of Lagrange’s multipliers is an important technique applied to determine the local maxima and minima of a function of the form f (x, y, z) subject to equality constraints

[tex]& \nabla f=\lambda \nabla y_1+\mu \nabla g_2 \\[/tex]

[tex]\Rightarrow & \langle 2 x, 2 y, 2 z\rangle=\lambda\langle 2 x, 2 y, 0\rangle+\mu\langle 1,1,1\rangle \\[/tex]

[tex]\Rightarrow & \langle 2 x, 2 y, 2 z\rangle=\langle 2 x \lambda+\mu, 2 y \lambda+\mu, \mu\rangle[/tex]

[tex]2 x & =2 x \lambda+\mu-(1) \\[/tex]

[tex]2 y & =2 y \lambda+\mu \ldots(2) \\[/tex]

[tex]2 z & =\mu \quad \ldots(3)[/tex]

From equation (1) [tex]$2 x(1-\lambda)=\mu=2 z$[/tex]

From equation (2) [tex]$2 y(1-\lambda)=\mu=2 z$[/tex]

Therefore if [tex]\lambda=1$[/tex], then z = 0

if [tex]\lambda \neq 1, \quad x=y=\frac{2}{1-\lambda}[/tex]

if z = 0, then after solving.

[tex]x^2+y^2=1$\\ $2 x+y=1$[/tex]

We get two points: [tex]${(1,0,0)} \geq {(0,1,0)}$[/tex]

and if x = y.

Then:

[tex]& x^2+y^2=1 \Rightarrow x^2+x^2=1 \Rightarrow 2 x^2=1 \Rightarrow x=\pm \sqrt{\frac{1}{2}} \\ & x+y+z=1[/tex]

x + y + z = 1

[tex]\Rightarrow[/tex] x + x + z = 1

[tex]\Rightarrow[/tex] 2 = 1 - 2x

= [tex]1-2\left(\pm \sqrt{\frac{1}{2}}\right)=1 \pm \sqrt{2}[/tex]

Therefore the corresponding points on the ellipse.

[tex]P_1\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 1-\sqrt{2}\right) \& P_2\left(-\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 1+\sqrt{2}\right)[/tex]

Now we have four points

Closest to the origin:

1) (1, 0, 0)

2) (0, 1, 0)

The farthest to origin:

3) [tex]$\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 1-\sqrt{2}\right)[/tex]

4) [tex]$\left(-\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}, 1+\sqrt{2}\right)[/tex]

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Answer:225 degrees

Step-by-step explanation:

Answer: A

A. y=-4x-2. Two lines are parallel if they have the same slope. In this case, the line given by y = -4x + 3 has a slope of -4, so the equation of the line that is parallel to it must also have a slope of -4. The equation y = -4x - 2 has a slope of -4, so it is the equation of a line that is parallel to the line given by y = -4x + 3.

The y-intercept of the line having point (3,−1) and slope -5 is 14.

The equation of a line in the slope-intercept form is:-

y = mx+c

where 'm' is the slope and 'c' is the y-intercept.

here, m= -5

∴ y = -5x + c  .........(Partial equation)

to find b substitute (3,-1) into the partial equation

where,

x= 3 and y= -1 by substituting the values,

-1 = (-5)×3 + c => c = 15 - 1

∵ c = 14

Thus, the value of the y-intercept is 14.

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The correct question is:-

Daliyah is given one point on a line as (3,−1) and the slope of the line as −5. what is the y-intercept of the line?

Answer:

Step-by-step explanation:

Here we have the second derivative of a function and we need to go back to the original function. This situation is inverse of taking derivative which is called antiderivative or integral. But because we have the second derivative, we have to integrate twice because the first integral will give me the first derivative then the second integral will give me the function itself.

1. Let's start taking the 1st integral of x-2 which is x²/2-2x+C1

2. Take the 2nd integral which is the integral of the 1st one

x³/6-2x²/2+C1x+C2

remember: the integral is the inverse of derivative, that means we have to add 1 to the power of x and divide by that power. I did that in steps 1 and 2.

3. After simplifying the second term, f(x)=x³/6-x²+C1x+C2 this is the function, but I have to find C1 and C2 the constants. We can use the two conditions given. the 1st one means that when x=1 , y=0

0=1/6-1+C1+C2 now combine like terms and move them to the left side of the equation C1+C2=5/6

the 2nd condition means that when x=3 , y=0

0=27/6-9+3C1+C2 after simplifying 3C1+C2=9/2

4. We got 2 equations with 2 variables C1 and C2, we can solve these system of equations. You can eliminate C2 and find C1 which is 11/6 then plug in C1 in any of the two equations to find C2, which is -1

5. Last step substitute C1 and C2 in the function from step 3 and you will get f(x)=x³/6-x²+11x/6-1

The wind speed of a hurricane with an air pressure of 956 kPa will be 2289.4 knots.

Given is that relationship between the air pressure inside a hurricane and its maximum sustained wind speed → y = 1.15x + 1190, where [x] is the air pressure in millibars (kPa) and [y] is the wind speed in knots (nautical miles per hour).

The air pressure inside a hurricane and its maximum sustained wind speed are related as → y = 1.15x + 1190

For x = 956 kpa of pressure, we can write -

y = 1.15 x 956 + 1190

y = 2289.4 knots

Therefore, the wind speed of a hurricane with an air pressure of 956 kPa will be 2289.4 knots.

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Answer:

Two sets of dimensions are possible:

              L(m)   W(m)   Area(m^2)   Fence (m) Perimeter

Set 1         19        16       304              51

Set 2         32      9.5      304              51

Step-by-step explanation:

Let's set L and W for the Length and Width of the perimeter.  We know that:

Area = L*W

Area = 304 m^2

Perimeter = 51 m

Let's say that the side bounded by the river is the length side.   No fence is required for that portion, so the fencing for the perimeter is given by:

Perimeter, P = L + 2W

P = 51 m

We can write:   L+2W=51 m

Rearrange to isolate one of the two variables.  Lets pick L:

 L+2W=51

 L=51-2W

L = (51-2W)

Now use this in the area equation:

304 m^2 = L*W

304 m^2 = (51-2W)*W

304 m^2 = (51-2W)*W

304 m^2 = 51W-2W^2

Arrange this in the standard form of a quadratice equation and solve:

-2W^2+51W-304 m^2 = 0

W = 16 and 9.5 meters

W = 16 m

Since L=51-2W

 L=51-2*(16)

L = 51-32

L = 19 m

Area = L*W

L*W = 304 m^2

(19m)W = 304 m^2

W =  16 m

Check:

Does a length (L) of 19 m and a width (W) of 16 m give an area of 304 m^2 and a fencing perimeter of 51 m if one of the L sides is the river?

Area = (19 m)*(16 m) = 304 m^2  YES

Per.  = 2*(16)+ 19 = 51 m  YES

W = 9.5 m

Ding the same calculation for a width of 9.5 m also satisfies all the equations, as per above.  

Both values of W are valid, so the twos sets of possible dimensions are:

                   L(m)    W(m)   Area(m^2)   Fence (m) Perimeter

Set 1          19      16       304              51

Set 2          32      9.5      304              51

The area of the floor of the scale drawing and the scale factor are 72 in² and 1/2 respectively.

Actual Length = 28 feets

Actual width =  38 feets

Model width = 14 inches

The scale factor = 14/28 =1/2

The length of the model can  be calculated thus :

Actual Length × scale factor

Model width = 38 × 1/2= 19 inches

The Area of a rectangle can be calculated using the relation :

Area = Length × width

Area of floor = 19 inches × 14 inches = 266 inch²

Therefore, the area of the floor of the scale drawing is 266 in²

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The sample with more observations and/or more variability in the data will generally have a larger chi-square test statistic.

Chi-square test statistic is a measure of how well a set of observed data fits a theoretical model. The larger the sample size and the more variability in the data, the more likely it is that the theoretical model won't fit the data perfectly. Therefore, the larger sample size and variability in the data will lead to a larger chi-square test statistic.

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Answer:

16 is the third quartile

Answer:

To determine how long it would take a 12-inch candle to burn down, we need to first determine the relationship between the length of the candle and the time it takes to burn down. Since the candles are the same thickness and make, we can say that the time it takes for a candle to burn down is directly proportional to its length. This means that we can write an equation to represent the relationship between the length of the candle and the time it takes to burn down as follows:

t = k * l

where t is the time it takes for the candle to burn down, k is the proportionality constant, and l is the length of the candle.

We can use the information given in the problem to find the value of k. We know that an 8-inch candle burns down in 4 hours, so we can substitute these values into the equation above to find the value of k, as follows:

4 = k * 8

Solving for k, we get:

k = 0.5

Now that we know the value of k, we can use the equation above to determine how long it would take a 12-inch candle to burn down. Substituting the values for t, k, and l into the equation, we get:

t = (0.5) * (12) = 6

This means that it would take a 12-inch candle 6 hours to burn down.

The range of rate that Nadia can type is 27 to 37.

We will calculate the lower and upper limit stating the minimum and maximum number of words Nadia can type in a minute. It will depict the range of rate.

Minimum number of words = 32 - 5

Performing subtraction on Right Hand Side of the equation

Minimum number of words = 27

Maximum number of words = 32 + 5

Performing subtraction on Right Hand Side of the equation

Minimum number of words = 37

Hence, Nadia's range of word typing is 27 to 37 per minute.

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If the value of the base is y, then y^(3x+2) = 5x(2x-4)

logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if b^x= n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log8 base 2.

Also if 10² = 100 , then 2 = log100 base 10.

representing the unknown base to be y

then, log 5x(2x-4) base y = 3x+2

from the law of logarithm

log a base b = x, then b = a^x

Similarly log 5x(2x-4) base y = 3x+2

then 5x(2x-4) = y^(3x+2)

Therefore the exponential equation of the logarithm is 5x(2x-4) = y^(3x+2)

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The required measure of length of the box is 2.9 inches. Option A is correct.

Volume is defined as the mass of the object per unit density while for geometry it is calculated as profile area multiplied by the length at which that profile is extruded.

Here,
according to the question,
h = 3l - 2.5
w = 3/4l
The volume of the box,
40 = l × w × h
40  = l × 3/4l × [3l-2.5]
40 = 3/4l²[3l - 2.5]
40 = 9/4l³ - 7.5/4l²

9/4l³ - 7.5/4l² - 40 = 0
t = 2.91932
t ≈ 2.9 inches

Thus, the length of the box must be at least 2.9 inches. Option A is the right answer.

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There are no free variables for the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution.

For the stated question-

It is necessary to know that a system is consistent (i.e., there is no pivot in the last column of the augmented matrix) and that there are no free variables in order to determine whether it has a unique solution (like a pivot position in every column of the coefficient matrix).

Also take note that this rules out having fewer rows than columns.

Thus, there are no free variables for the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution.

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i just got some free money off of you kid

Answer:

f(x)=-2x+5

f(x)=3

then,

f(x)=-2x+5

or, 3=-2x+5

or, 2x=5-3

or,2x=2

or, x=2÷2

x=1

so, the value of x is 1

Answer:

x=1

Step-by-step explanation:

3= -2x+5

-5.       -5

-2= -2x

/-2. /-2

1=x

hopes this helps

Answer:

A. 13/15​

B. 13/10​

C. 22/21

D. ​−3/20​

Step-by-step explanation:

Just make the denominators equal to each other and solve. Ask your teacher for extra help

Sum of multiplied differences= -962.676 and β1 = -962.676 / ((-18.381)^2 + (22.619)^2 + (-13.381)^2 + (-10.381)^2 + (10.619)^2 + (2.619)^2 + (28.619)^2 + (-17.381)^2 + (-17.381)^2 + (-9.381)^2 + (0.619).

To estimate the parameters of the model, we can use the following steps:

Calculate the mean of the first test scores and the mean of the second test scores.

Calculate the difference between each student's first test score and the mean first test score, and the difference between each student's second test score and the mean second test score.

Multiply the differences from step 2 together for each student, and sum the results.

Divide the sum from step 3 by the sum of the squares of the differences from step 2. This will give us the value of β1.

Calculate β0 by substituting the values of β1 and the mean first and second test scores into the equation: Second Test Grade = β0 + β1(First Test Grade)

Using these steps, we can calculate the estimates of the parameters as follows:

Mean first test score = (50+91+55+58+79+70+97+51+51+59+69+56+56+41+49+58+58+99+87+95+97)/21 = 68.381

Mean second test score = (70+58+74+65+60+60+48+73+73+66+67+70+73+76+72+68+73+53+58+50+55)/21 = 64.762

Difference between student 1's first test score and mean first test score = 50 - 68.381 = -18.381

Difference between student 1's second test score and mean second test score = 70 - 64.762 = 5.238

Difference between student 2's first test score and mean first test score = 91 - 68.381 = 22.619

Difference between student 2's second test score and mean second test score = 58 - 64.762 = -6.762

Sum of multiplied differences = (-18.3815.238) + (22.619-6.762) + (-13.3818.238) + (-10.381-0.762) + (10.619*-6.238) + (2.619*-7.238) + (28.619*-16.762) + (-17.3819.238) + (-17.3819.238) + (-9.381*-0.762) + (0.619*-0.238) + (-2.3816.238) + (-2.3817.238) + (-27.38116.238) + (-18.3813.238) + (-10.381*-3.762) + (-10.3815.238) + (-10.3815.238) + (30.619*-11.762) + (18.619*-6.762) + (26.619*-14.762) + (28.619*-9.762) = -962.676

β1 = -962.676 / ((-18.381)^2 + (22.619)^2 + (-13.381)^2 + (-10.381)^2 + (10.619)^2 + (2.619)^2 + (28.619)^2 + (-17.381)^2 + (-17.381)^2 + (-9.381)^2 + (0.619)

Thus, Sum of multiplied differences= -962.676 and β1 = -962.676 / ((-18.381)^2 + (22.619)^2 + (-13.381)^2 + (-10.381)^2 + (10.619)^2 + (2.619)^2 + (28.619)^2 + (-17.381)^2 + (-17.381)^2 + (-9.381)^2 + (0.619).

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Step-by-step explanation:

So 2 * 4 = 8

8 * 4 = 32

32 * 4 = 128

128 * 4 = 512

512*4=2048

2048*4=8192

8192*4=32768

32768*4=131072

131072*4=524288

524288*4=2097152

2097152*4=8388608

8388608*4=33554432 and that would probably be your final answer

The total spent at the mall by Gabe is $72.84 and the expression which represents the amount is 12.92 k + 27 + 20 or 12.92 k + 47.

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement.

Given:

12.92 k + 27, where k is the number of items bought,

Calculate the  expression which represents the amount Gabe spent at the mall as shown below,

Total amount = Amount spent in one store + Amount spent in the second store

Total amount = 12.92 k + 27 + 20

Calculate the estimate of Gabe spent by putting  k = 2,

Total spent in mall = 12.92 × 2 + 27 + 20

Total spent in mall = 25.84 + 47

Total spent in mall = $72.84

Therefore, the total spent at the mall by Gabe is $72.84, and the expression which represents the amount is 12.92 k + 27 + 20 or 12.92 k + 47.

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five out of of the twenty-five top-grossing films of 2017 in the united states were part of a movie franchise

If you were to attribute this film, that means you'd say that (something) caused the success of the movie Rocky. This question is basically asking what do you think made this movie a big hit? Personally, I'd say it's because of how the storyline and plot turned out. It's very intriguing and it doesn't just make the movie about boxing. The film adds some love between the protagonist and some other character, and not only that, but the movie teaches you something. Because Rocky lost to Apollo, Rocky didn't care. At least he still had Adrian. So, the movie teaches you that, even if you do lose, at least you still have something and/or someone with you.

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well, if the population is doubling from whatever they happen to be, so that means the growth rate is 100%, so if they're hmmm "g", then later they become "2g", or doubled, and the later becomes "4g" and so on, so the rate is simply 100%, because "g" plus "g" is just 2g, and "2g" plus "2g" is just 4g and so on, anyhow

[tex]\textit{Periodic/Cyclical Exponential Growth} \\\\ A=P(1 + r)^{\frac{t}{c}}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &5000\\ r=rate\to 100\%\to \frac{100}{100}\dotfill &1\\ t=hours\\ c=period\to &6 \end{cases} \\\\\\ A=5000(1 + 1)^{\frac{t}{6}}\implies A=5000(2)^{\frac{t}{6}}\implies {\Large \begin{array}{llll} P(t)=5000(2)^{\frac{t}{6}} \end{array}}[/tex]

The required function f when f"(x) = x^-2, x > 0, f(1) = 0, f(2) = 0 is f(x) = - ln x  + (ln 8) x/7 + (ln 8)/7

In tthis question we have been given the second derivative of function

f"(x) = x^-2, x > 0, f(1) = 0, f(2) = 0

We need to find a function f(x).

Consider f"(x) = x^-2

Integrating above expression with respect to x we get,

f'(x) = -1/x + C

againg integrating above expression with respect to x we get,

f(x) = - ln x + Cx + D

Now we need to find C and D.

Given that f(1) = 0

- ln (1) + C + D = 0     

C + D = 0                       ........... (1)

And f(8) = 0

- ln 8 + 8C + D = 0   

8C + D = ln 8                ..............(2)

subtracting equation (1) from equation (2),

C = (ln 8)/7

Subtituting above value of C in equation (1) we get,

D = -(ln 8)/7

So, f(x) = - ln x  + (ln 8) x/7 + (ln 8)/7

Therefore, the required function : f(x) = - ln x  + (ln 8) x/7 + (ln 8)/7

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The sequence of the energy conversion is;  Kinetic → electrical → light Option A

From the first law of thermodynamics, we know that energy can neither be created nor destroyed but it can be converted from one form to the other. We have been told that in this case, the flashlight shown below has no batteries. Instead, it is operated by squeezing and letting go of the handle.

The letting go of the handle is kinetic energy which would lead to electrical energy that is then converted into the visible light energy.

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The value that 50% of the data, as shown in the box plot, lie above and what it is called is: B. 50.5, the median

A box plot that displays the data set has the value of the median indicated at the point where a vertical line divides the rectangular box.

The median can be referred to as the second quartile (Q2) or 50th percentile of the data set.

The median value shows that 50% of the data set is above it while 50% is below it also.

The box plot in the given image shows that the vertical line crosses the rectangular box at 50.5.

50.5 is therefore the median, and this means that 50% of the data lie above 50.5.

Therefore, the answer is:

B. 50.5, the median

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