Consider a cookie recipe in which 1½ cups of chocolate chips are needed to make 25 cookies. Lara wants to double the number of cookies. How many cups of chocolate chips should she use?

Answer:

[tex]\large\text{$y=ke^{\frac{1}{3}x^3+x}$}[/tex]

Step-by-step explanation:

Given differential equation:

[tex]\large\text{$\dfrac{\text{d}y}{\text{d}x}=y(x^2+1)$}[/tex]

Rearrange the equation so that all the terms containing y are on the left side, and all the terms containing x are on the right side:

[tex]\large\text{$\implies \dfrac{1}{y}\;\text{d}y=(x^2+1)\;\text{d}x$}[/tex]

Integrate both sides, remembering to add the constant of integration (C):

[tex]\large\begin{aligned}\implies \displaystyle \int\dfrac{1}{y}\;\text{d}y & =\int(x^2+1)\;\text{d}x\\\ln y & = \dfrac{1}{3}x^3+x+\text{C}\end{aligned}[/tex]

Rewrite C as ln k:

[tex]\large\text{$\implies \ln y = \dfrac{1}{3}x^3+x+\ln k$}[/tex]

Solve for y, applying:

[tex]\large\text{$ \implies e^{\ln y} =e^{\frac{1}{3}x^3+x+\ln k}$}[/tex]

[tex]\large\text{$ \implies y=e^{\ln k} \cdot e^{\frac{1}{3}x^3+x}$}[/tex]

[tex]\large\text{$\implies y=ke^{\frac{1}{3}x^3+x}$}[/tex]

Integration rules

[tex]\boxed{\begin{minipage}{3.6 cm}\underline{Integrating $\frac{1}{x}$}\\\\$\displaystyle \int \dfrac{1}{x}\:\text{d}x=\ln |x|+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{3.6 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5.1 cm}\underline{Integrating a constant}\\\\$\displaystyle \int n\:\text{d}x=nx+\text{C}$\\\\(where $n$ is any constant value) \end{minipage}}[/tex]

Answer:

[tex]{ \tt{ \frac{dy}{dx} = y( {x}^{2} + 1) }} \\ [/tex]

- Simplify by collecting each term according to its corresponding d

[tex]{ \tt{ \frac{dy}{y} = ( {x}^{2} + 1) \: dx}} \\ [/tex]

- Integrate both sides;

[tex]{ \tt{ \int \frac{1}{y} \: dy = \int ( {x}^{2} + 1) \: dx }} \\ \\ { \tt{ ln(y) = \frac{1}{3} {x}^{3} + x + c }}[/tex]

- To make y the subject, you must remove the natural log;

[tex]{ \tt{ log_{e}(y) = \frac{1}{3} {x}^{3} + x + c }} \\ \\ { \tt{y = {e}^{( \frac{1}{3} {x}^{3} + x + c) } }} \\ [/tex]

The scale from a playground to this drawing is 5 ft to 1 cm. The scale from the same playground to another drawing is 8 ft to 1 cm. What are the side lengths of the playground in the other scale drawing?

The side lengths of the playground in the other scale drawing is [tex]\frac{5a}{8}[/tex].

By using geometry we answer this question.

Geometry is nothing but  the branch of mathematics which deals with measurements of shapes.

Here, we take the scale from a playground to this drawing as "a" and The scale from the same playground to another drawing as "b".

First, we see the scale from a playground to this drawing.

[tex]\frac{a}{l}= \frac{1}{5}[/tex]

[tex]l=5a[/tex]

Now we see the scale from the same playground to another drawing.

[tex]\frac{b}{l} =\frac{1}{8}[/tex]

[tex]b=\frac{l}{8}\\[/tex]

Here, l=5a then

[tex]b=\frac{5a}{8}[/tex]

Therefore, the side lengths of the playground in the other scale drawing is [tex]\frac{5a}{8}[/tex].

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Answer:

Slope = (-7/4)

Step-by-step explanation:

Point 1: (-3, 4); Point 2: (1, -3)

            (x₁, y₁)               (x₂, y₂)

       y₂ - y₁        -3 - 4           -7           -7

m = ----------- = ----------- = ----------- = ------

       x₂ - x₁         1 - (-3)         1 + 3        4

I hope this helps!

The equation of the composite function h(x) is h(x) = -10x^2 + 40x

The functions are given as

f(x) = 2x – 8

g(x) = –5x

The equation of the composite function is given as

h(x) = f(x)g(x)

Substitute f(x) = 2x – 8 and g(x) = –5x in the equation h(x) = f(x)g(x)

So, we have

h(x) = (2x - 8) x (-5x)

Evaluate the product

h(x) = -10x^2 + 40x

Hence, the equation of the composite function h(x) is h(x) = -10x^2 + 40x

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The yearly interest rate percentage is 7%

The rate of interest can be found dividing the interest and  the product of the principal and time period which can be written as

R = I / PT

where R is the Rate of interest

          I is the Interest

          P is the Principal amount

          T is the time period.

Given values are

P = $5,875

 I = $1,675

 T = 4 years

Then,

R = I/PT

   = 1,645/(5,875 x 4)

   = 1645 / 23500

   = 0.07

The Yearly interest rate percentage is

 =0.07 x 100

 = 7%

Therefore, the yearly rate percentage is 7%

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The function which passes from (1,4) and (5,1) is y= -7/20x² + 27/20x   +3

and the net change between the indicated points is -3, whereas the average rate change is -3/4.

Given, from the graph, the function passes (1,4) and (5,1).

Let y = ax² + bx + 3

from point 1.

the equation is: a + b+ 3 = 4 eq(1)

from point 2.

the equation is 25a + 5b + 3 = 1 eq(2)

solving equation 1 and equation 2.

we get, 20b = 99-72

20b = 27

b = 27/20

substitute b value in equation 1.

a + 27/20 + 3 = 4

20a = -7

a = -7/20

hence y = -7/20x² + 27/20x +3

(a) net change = change in y coordinates.

net change = y₂ - y₁

net change = 1 - 4

= -3

(b) Average rate = change in y coordinates / change in x coordinates

= 1 - 4/5-1

= -3/4

hence  we get the required answers.

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Answer:

Step-by-step explanation:

go

Answer:

4 - 3x

Step-by-step explanation:

please mark me as brainliest

Answer:

3x-4= Y

Y= her dad's age.

After rounding, we can say that there are about 1,140 candles in the 22 packets.

First let's get the exact number, and then round it as we need to do.

We know that each packet contains 52 candles, and there are 22 packets, so the total number of candles is given by the product between these two values, which gives:

C = 52*22 = 1,144 candles.

Now we want to round it to the nearest ten, to do so, we need to look at the digit at the right of the tens unit (in this case the ones digit) and see, if it is 5 or more, we round up, if it is 4 or less, we round down.

We can see that it is a 4, so we round down here:

C = 1,140

There are about 1,140 candles in the 22 packets.

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[tex]\displaystyle \lim_{x\to 44}~\cfrac{\sqrt{x+5}-7}{x-44}\hspace{5em}\stackrel{\textit{L'Hopital's rule}}{\lim_{x\to 44}~\cfrac{ ~~ \frac{d}{dx}[\sqrt{x+5}-7] ~~ }{\frac{d}{dx}[x-44]}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\cfrac{d}{dx}[\sqrt{x+5}-7]\implies \cfrac{1}{2}(x+5)^{-\frac{1}{2}}(1)\implies \cfrac{1}{2\sqrt{x+5}} \\\\\\ \cfrac{d}{dx}[x-44]\implies 1 \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \lim_{x\to 44}~\cfrac{\sqrt{x+5}-7}{x-44}\implies \lim_{x\to 44}~\cfrac{ ~~ \frac{1}{2\sqrt{x+5}} ~~ }{1}\implies \lim_{x\to 44}~\cfrac{1}{2\sqrt{x+5}}\implies \cfrac{1}{2\sqrt{44+5}} \\\\\\ \cfrac{1}{2\sqrt{49}}\implies \cfrac{1}{2(7)}\implies \cfrac{1}{14}[/tex]

Answer:

The mean will move to the left as more affordable vehicles are introduced.

Step-by-step explanation:

It is given that a car salesman sells cars with prices ranging from $5,000 to $45,000. The histogram shows the distribution of the numbers of cars he expects to sell over the next 10 years.

We need to find how will the distribution be affected if he decides to also deal in cars that cost less than $5,000 and projects selling 200 of them over the next 10 years.

Now, let us understand what is meant by mean;

What is the Mean ?

The mean of a set of data is its average value.

Further, It is given that the car salesman sells cars with prices ranging from $5,000 to $45,000.

Additionally, the salesperson has noticed that a lot of students are searching for vehicles around $5,000.

If he chooses to sell vehicles as well and anticipates selling 200 of them over the following ten years,

The histogram will deviate from its mean if cards with values less than $5,000 are also included in the data set.

The mean will move to the left as more affordable vehicles are introduced.

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Answer:

Step-by-step explanation: 6.140

Answer:

only D is negative on the leading term

Step-by-step explanation:

even roots bounce at (0,0)

negative leading coefficient: initial graph descending to right, curve up

4 local extremes means remaining 5 degree of the function

[tex] \bf\longrightarrow \large\frac{ \frac{ \frac{a( {r}^{4} - 1)}{ \cancel{r - 1}} }{a( {r}^{2} - 1) } }{ \cancel{r - 1}} = 10 \\ [/tex]

[tex]\bf\longrightarrow {r}^{4} - 1 = {10r}^{2} - 10[/tex]

[tex]\bf\longrightarrow {r}^{4} - {10r}^{2} + 9 = 0[/tex]

[tex]\bf\longrightarrow \: {a}^{2} - 10 {a}^{2} + 9 = 0[/tex]

[tex]\bf\longrightarrow \: a = \frac{10± \sqrt{100 - 36} }{2} = \frac{10±8}{2} [/tex]

r = 3 or r = 1 but r ≠ 1

[tex]\therefore[/tex] r = 3

[tex]\bf\longrightarrow \: a_3 = 54 \\ \bf\longrightarrow \: a {r}^{2} = 54 \\ \bf\longrightarrow \: a = \frac{54}{9} = 6[/tex]

Answer:

y = 2/3x + 9

Step-by-step explanation:

The equation of a line is defined as y = mx + b

with m being the slope, and b being he intercept.

So to figure out the equation for the line with a slope of 2/3 and an intercept of 9, we can just plug those values into our line equation!

So take y = (slope)*x + *(intercept)

and plug your values in and you get...

y = 2/3 * x + 9

Answer: the answers for the quiz are

1. What is the slope (rate of change) of the line in the graph show below? Answer is A. -2

2. Find the slope (rate of change) of a line that passes through (-2, -3) and (1, 1). Answer is D. 4/3

3. constant of variation of -4y=8x  answer is B. -2

4 the value of y when x=10 answer is B. 30

5. the equation for the line with slope 2/3 and y-intercept 9 answer is C. y=2/3x+9

6 is the equation in slope intercept form for the line that passes through the points (1, -3) and (3, 1) answer is D. y=2x-5

Step-by-step explanation:

Using proportions, it is found that the cat has to go 9/16 further to reach the gate.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct(when both increase or both decrease) or inverse proportional(when one increases and the other decreases, or vice versa), can be built to find the desired measures in the problem, or equations to find these measures.

An entire distance is represented by the proportion of 1 = 100%. The cat has gone 7/16 of the distance, hence the fraction corresponding to the remaining distance that he has to go is given by:

1 - 7/16 = 16/16 - 7/16 = 9/16.

Hence the cat has to go 9/16 further to reach the gate.

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Answer: Here's the answer step by step

Step-by-step explanation: First, find the square root of 1000000 base to and leave your answer in base two

find the square root of 111 in base 2

simplify 342+134-233 in base 5

divide 100001 by 11 in base 2

convert 123.12 in base 3 to a number in base 10 and make sure you leave your answer in base 2

Last step: convert 3.875 in base 10 to a number in base 2

Answer:

Step-by-step explanation:

You want the perimeter of each of the figures defined by the coordinates of their vertices. You are told to use the distance formula as necessary.

The first attachment shows the graph of the triangle. The distance formula is needed only for the length of the diagonal segment:

  d = √((x2 -x1)² +(y2 -y1)²)

  d = √((3 -(-2))² +(-3 -3)²) = √(5² +(-6)²) = √61 ≈ 7.81

The lengths of the horizontal and vertical legs of the triangle are the difference of their x- and y-coordinates, respectively.

  CB = 3 -(-2) = 5

  CA = 3 -(-3) = 6

The perimeter is the sum of the side lengths:

  P = CA +CB +AB = 6 +5 +7.81 = 18.81

The perimeter of the triangle is 18.81 units.

The second attachment shows the graph of the parallelogram. As with the triangle, we only need to use the distance formula for the length of the diagonal side. Here is the length of ML.

  d = √((4 -2)² +(1 -(-2))²) = √(2² +3²) = √13 ≈ 3.606

The length of the horizontal legs is the difference of their x-coordinates.

  KL = 4 -(-1) = 5

Opposite sides are congruent, so the perimeter is double the length of two adjacent sides.

  P = 2(3.606 +5) ≈ 17.21

The perimeter of the parallelogram is about 17.21 units.

Answer:

1.  18.81 units

2.  17.21 units

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7.3 cm}\underline{Distance between two points}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

Given vertices of ΔABC:

Plot the vertices on the given graph paper and join with line segments to create the triangle.

As points B and C share the same y-coordinate:

[tex]\implies BC = |x_B-x_C|=|3-(-2)|=5\:\: \sf units[/tex]

As points A and C share the same x-coordinate:

[tex]\implies AC = |y_A-y_C|=|3-(-3)|=6\:\: \sf units[/tex]

Use the distance formula to find the length AB:

[tex]\begin{aligned}AB & =\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}\\& =\sqrt{(3-(-2))^2+(-3-3)^2}\\& =\sqrt{(5)^2+(-6)^2}\\& =\sqrt{25+36}\\& =\sqrt{61}\\\end{aligned}[/tex]

The perimeter of a two-dimensional shape is the distance all the way around the outside.

[tex]\begin{aligned}\textsf{Perimeter of $ABC$} & = AB + BC + AC\\& = \sqrt{61}+5+6\\& = 11+\sqrt{61}\\& = 18.81\:\: \sf units\:(nearest\:hundredth)\end{aligned}[/tex]

Given vertices of polygon KLMN:

Plot the vertices on the given graph paper and join with line segments to create the polygon.

As the y-coordinate of points K and L, and M and N are the same, KL and MN are parallel line segments.

As the difference between the x-coordinates of K and N, and L and M is 2 units, KN and LM are parallel line segments.

Therefore, the polygon is a parallelogram.

A parallelogram has two pairs of opposite sides that are equal in length.

Therefore, KL = NM and KN = LM.

As points K and L share the same y-coordinate:

[tex]\implies KL = |x_K-x_L|=|-1-4|=5\:\: \sf units[/tex]

Use the distance formula to find the length KN:

[tex]\begin{aligned}KN & =\sqrt{(x_N-x_K)^2+(y_N-y_K)^2}\\& =\sqrt{(-3-(-1))^2+(-2-1)^2}\\& =\sqrt{(-2)^2+(-3)^2}\\& =\sqrt{4+9}\\& =\sqrt{13}\\\end{aligned}[/tex]

The perimeter of a two-dimensional shape is the distance all the way around the outside.

[tex]\begin{aligned}\textsf{Perimeter of $KLMN$} & = 2\:KL + 2 \:KN\\& = 2 \cdot 5 + 2\cdot \sqrt{13}\\& =10 + 2\sqrt{13}\\& = 17.21\:\: \sf units\:(nearest\:hundredth)\end{aligned}[/tex]

Answer:

2, 4 6, 8

Step-by-step explanation:

The counting numbers are the set of numbers: 1,2,3,4,... It follows that pattern forever.  No, 1/2 or .0745.

Answer:

2 4 6 8 is the answer for your question

Step-by-step explanation:

when a line intersects with 2 (or more) parallel lines, the angles at the intersection points are the same for each parallel line.

therefore,

5 + 10x = 85

10x = 80

x = 8

so, B is the right answer option.

√80 on the number line would be marked at (8.94, 0) on the number line.

Hope this graph helps!

Answer:

28.26 inches^2

Step-by-step explanation:

The area of a circle is given by the formula [tex]A = \pi r^2[/tex].
We know that the diameter of the circle is 6 inches, therefore, its radius 3 inches (the radius is exactly one half of the circle's diameter).

Substituting into the formula

[tex]A = \pi r^2 \\ \\\to A = \pi \times 3^2 \\ \\\to A = \pi \times 9 = 28.26[/tex]

The value of x is 13

From the diagram shown, we have that the line segments as;

NK = NM + ML + LK

Where;

Substitute the values into the equation

(x - 6) + 9 + 2x - 19 = 23

collect like terms

x + 2x -6 + 9 - 19 = 23

Add like terms

3x - 16 = 23

3x = 23 + 16

3x = 39

Divide both sides by 3

x = 39/ 3

x = 13

Thus, the value of x is 13

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Answer:

Step-by-step explanation:

                           [tex]\displaystyle\\\boxed {M_{AB}=(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2} ) }[/tex]

1) E(-5,-3)      F(3,7)      M(x,y)=?

[tex]\displaystyle\\M(x,y)=(\frac{-5+3}{2} ),\frac{-3+7}{2})\\\\ M(x,y)=(\frac{-2}{2},\frac{4}{2} )\\\\ M(x,y)=(-1,2)\\\\Thus, \ M(-1,2)[/tex]

2) L(-8,11)     N(-3,12)      M(x,y)=?

[tex]\displaystyle\\M(x,y)=(\frac{-8+(-3)}{2} ,\frac{11+12}{2} )\\\\M(x,y)=(\frac{-11}{2},\frac{23}{2})\\\\ M(x,y)= (-5.5,11.5)\\\\Thus, M(-5.5,11.5)[/tex]

3) A(-4,6)    B(2,4)    M(x,y)=?

[tex]\displaystyle\\M(x,y)=(\frac{-4+2}{2},\frac{6+4}{2})\\\\ M(x,y)=(\frac{-2}{2},\frac{10}{2})\\\\ M(x,y)=(-1,5)\\\\ Thus,\ M(-1,5)[/tex]

Answer:

C. =

Step-by-step explanation:

This is because 18% is 18/100. You could find the answer one of two ways. Divide by two or multiply by 2.

The value of the given functions are:

(i) f + g = x² + x - 6

(ii) f - g = -x² + x - 4

(iii) f . g = x³ - 5x² - x + 5

(iv) f/g = ( x - 5)/(x +1)(x - 1)

(v) g(f(x)) = x² - 10x + 24

The two functions are given that

f(x) = x - 5

g(x) = x² - 1

We have to find

(i) f + g

f(x) + g(x) = ( x-5)  + (x² - 1)

                = x² + x -5 -1

               = x² + x -6

(ii) f - g

f(x) - g(x) = (x-5)- (x² - 1)

              = -x² + x -5 +1

              = -x² + x -4

(iii) f . g

f(x) . g(x) = (x-5)(x²-1)

              =x(x² - 1) -5(x² - 1)

              = x³ - x - 5x² + 5

              = x³ - 5x² - x + 5

(iv) f/g

f(x) / g(x) = (x-5) / ( x² - 1)

              = (x - 5) / (x+1)(x-1)

(v) g(f(x)) = - 1 + ( -5 + x )²

              = -1 + 25 + x² - 10x

              = x² - 10x + 24

Therefore we get the value of the function, (i) f + g = x² + x - 6, (ii) f - g = -x² + x - 4,(iii) f . g = x³ - 5x² - x + 5, (iv)f/g = ( x - 5)/(x +1)(x - 1), (v) g(f(x)) = x² - 10x + 24.

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The complete question is:

F(x)=x-5 and g(x)=x2 - 1 find (i) f + g , (ii) f-g, (iii) f. g (iv) f/g (v) g(f(x)).

The inequality using interval notation are (-7, -4).

An interval is represented on a number line using interval notation.

For the given expression;

-7 < x < -4

As, x > -7; x is not equal to -7.

x < -4; here also x is not equal to -4.

So, the is a condition of open interval where both vales are not taken.

Thus, the intervals are written as; (-7, -4).

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