Finding the derivatives using The Chain Rule

The area of a circle is 12πx-60π unit².

By examining the given figure, there are two circles i.e the small circular hole and the bigger circle. Since the small is a hole then we have to subtract its area from the area of the bigger circle.

Given:

radius of small circle(r₂) = x+2

radius of bigger circle(r₁) = x+8

The formula of Area(A) = πr²

Area of the figure = A₁ - A₂

                             =  πr₁² - πr₂²

                             =  π(x+8)²- π(x+2)²

                             = π(x²+16x+64) -  π(x²+4x+4)

                             =  x²π+16πx+64π - x²π-4πx-4π

                            =   12πx-60π unit squared

Therefore,  the area of the circle is 12πx-60π unit squared. So the 2nd option is the answer.

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Answer:

Step-by-step explanation:[tex]x2{12}5\\ =18[/tex]

Answer:

4

Step-by-step explanation:

12÷3=4

Answer:

15/x

Step-by-step explanation:

x = fish tank

The point slope form of the equation is = P2 = (11, 2)

acc to our question-

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Directly proportional to x.

Acc to our question-

Hence,Directly proportional to x.

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Jeanne can make 3 pairs of shorts with 3 7/8 yards of fabric.

What do you mean by ratio?

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

For example, a/b, where a will be the numerator and b will be the denominator.

Proportion is the relation of a variable with another. It could be direct or inverse.

Given that,

The total amount of fabric = 3 7/8 yards = 31/8 yards.

Amount of fabric needed to make 1 pair of shorts = 1 1/4 = 5/4 yards.

The number of shorts that can be made = 31/8 ÷ 5/4

⇒ 3.1 so it will be 3 because no relevance of decimal shorts.

Hence "Jeanne can make 3 pairs of shorts with 3 7/8 yards of fabric".

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The equation of the line in slope intercept form is y = 6x + 5

The given parameters are

Slope, m = 6

Points, (x₁, y₁) = (-2, -7)

A linear equation can be represented as

y = slope * x + y-intercept

i.e.

y = m(x - x₁) + y₁

Substitute the known values in the above equation

So, we have the following equation

y = 6(x + 2) - 7

Expand

y = 6x + 12 - 7

Evaluate the like terms

y = 6x + 5

Hence. the linear equation is y = 6x + 5

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Answer:

1 carton

Step-by-step explanation:

3 (7/10)= 21/10

1/3 of 21/10= 7/10

7/10=1 carton

For each of the arithmetic sequences, we have that:

a)

b)

c)

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The nth term of an arithmetic sequence is given by the following rule:

[tex]a_n = a_1 + (n - 1)d[/tex]

In which [tex]a_1[/tex] is the first term of the sequence.

For the sequences in this problem, we have that:

Then the general rule is applied to find the nth term of the sequence, while the next two terms are found adding the common ratio.

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Answer:

241

Step-by-step explanation:

9 x 12 = 108 m (area of square)

19 x 14 = 266 / 2 = 133 (area of triangle)

108 + 133 = 241

Pairs (-1, 2.6) and (-1, -4.2), and (5.1, -3.3) and (5.1, 3.4) has distance greater than 6 unit. Options A and B are correct.

Given that,
The coordinate pairs separated by a vertical distance of greater than 6 units is to be determined.

Coordinate, is represented as the values on the x-axis and y-axis of the graph. while the coordinate x is called abscissa and the coordinate of the y is called ordinate.

here,
(A)
Since the x coordinate is the same in all the pairs, distance on the y axis can only be calculated.

(5.1, -3.3) and (5.1, 3.4)
y = 3.4 - (-3.3)
y = 6.7

Similarly distance,
(B) y = 6.8
(C) y = 5.6
(D) y = 3.5

Thus, Pairs (-1, 2.6) and (-1, -4.2), and (5.1, -3.3) and (5.1, 3.4) has distance greater than 6 unit. Options A and B are correct.

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The relationship between the number of weeks and cars are proportional since they are dependent on each other.

Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. Proportion is an equation that defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. In proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.

In this question, Alfred, buy three car each week. As the number of weeks increase, so is the number of his car increasing at definite proportions.

This implies that we can set an equation to represent the number of cars Alfred will have after certain number of weeks.

Let;

y = 3x

At week 10, Alfred will have

y = 3(10) = 30 cars

After 10 weeks, Alfred will have 30 cars

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The Missing Lengths are AB = 10 units, OP = 10 units and f = 4 units

Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent.

When two triangles ΔABC and ΔXYZ are similar then

                    AB/XY = BC/YZ = AC/XZ

Here we have

ΔABC and ΔOPQ which are similar and right-angled triangles

As we know in a right-angled triangle

=>  Hyp² = side² + side²  

From ΔABC,

AB² = AC² + BC²  

=> AB² = 6² + 4²

=> AB² = 100 = 10²

=> AB = 10 units

Since  ΔABC and ΔOPQ are similar

=>  AB/OP = BC/OQ = AC/PQ  

=>  10/OP = 4/ f = 6/ 6

Take 10/OP = 6/ 6  

=>  6(10) = 6(OP)

=>  OP = 10 units  

4/ f = 6/ 6  

=> f = 4 units    

Therefore,

The Missing Lengths are AB = 10 units, OP = 10 units and f = 4 units

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Answer:

v>3

Step-by-step explanation:

2v>7-1

2v>6

v>6/2

v>3

It takes Lena to compress the accordion completely and then stretch it back out to a length of 50cm.

It takes Lena 4.1 seconds to compress the accordion completely and then stretch it back out to a length  of 50cm.

I am assuming the function without a T: [tex]A(t) = 25_{cos}(t)+65[/tex]

The function is graphed in the figure attached, and the blue line is [tex]y = 50cm[/tex].

From the graph we see that the accordion is completely stretched  (has minimum length) at [tex]t = \pi[/tex] seconds, and stretches back  to 50cm at [tex]t = 4.069s[/tex]; therefore, it takes Lena 4.069 seconds to stretch the accordion back again, which to the nearest tenth of a second is  4.1s.

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Answer: 1.3s

Step-by-step explanation:

If you're solving for  [tex]A(t)=25cos( \pi t)+65\\[/tex]

Please find attached the graph of the image of ΔIJK following a translation transformation of (-17, 5), and a reflection across the line y = 5

A transformation is an operation that changes the location, size and shape of geometric figures.

The coordinates of the vertex of ΔIJK are; I(14, 8), J(8, 5), K(14, 2)

The translation transformation is 17 units left and 5 units up = T₍₋₁₇, ₅₎

The location of the vertices of the image of ΔIJK following the translation transformation are;

[tex]I(14,\, 8)\ \underset{\longrightarrow}{T_{(-17,\, 5)}}\ I'(-3,\, 13)[/tex]

[tex]J(8, 5)\ \underset{\longrightarrow}{T_{(-17,\, 5)}}\ j'(-9,\, 10)[/tex]

[tex]K(14, 2)\ \underset{\longrightarrow}{T_{(-17,\, 5)}}\ K'(-3,\, 7)[/tex]

The coordinates of a point following a reflection across the line y = 1 are as follows;

The coordinates of the vertices of the image I'J'K' relative to the line y = 1 are;

I'(-3, 13), Relative point I'(-3, 12)

J'(-9, 10) has a relative point at (-9, 9)

K'(-3, 7) has a relative point (-3, 6)

The coordinates of the vertices of the reflected image are therefore;

(x, y)  [tex]\underset{\longrightarrow}{R_{y = 1}}[/tex]  (x, -(y - 1))

Which gives;

I'(-3, 12)  [tex]\underset{\longrightarrow}{R_{y = 1}}[/tex] I''(-3, -12 + 1) = I''(-3, -11)

J'(-9, 9) [tex]\underset{\longrightarrow}{R_{y = 1}}[/tex] I''(-9, -9 + 1) = I''(-9, -8)

k'(-3, 6)  [tex]\underset{\longrightarrow}{R_{y = 1}}[/tex] k''(-3, -6 + 1) = k''(-3, -5)

Please find attached the image of the ΔIJK following the translation and reflection transformation

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At the end of the first round, the total time is 648 seconds ( option E )

An average is the result that you get when you add two or more numbers together and divide the total by the number of numbers you added together.

For example, if there are 10 students in a class with ages 10, 11, 12, 10, 10,12,12,10,11,12 their average age will be( 10+ 11+12+10+10+12+12+10+11+12)/10

for Mariam her average time is 76 which means her total time is 76× 2= 152

for Jake and Rafael their average time is 84 which means their total time is 84×4= 336

for Sam,his average time is 80 which means is total time is 80×2= 160

therefore the total time spent by the 4 students is 152+336+160=648seconds

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The equation of a line for y-intercept and the coordinate point is y=3x-1.

Given that, y-intercept is -1 and the line passes through the point (3, 8).

In Maths, an intercept is a point on the y-axis, through which the slope of the line passes. It is the y-coordinate of a point where a straight line or a curve intersects the y-axis. This is represented when we write the equation for a line, y = mx+c, where m is slope and c is the y-intercept.

Put, c=-1 and (x, y)=(3, 8) in y = mx+c, we get

8=m(3)-1

⇒ 3m=9

⇒ m=3

Substitute c=-1 and m=3 in y = mx+c, we get

y=3x-1

Therefore, the equation of a line for y-intercept and the coordinate point is y=3x-1.

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The greatest even integers that the 5 consecutive number has a sum of - 80 is - 12.

Even numbers are those numbers that can be divided into two equal groups or pairs and are exactly divisible by 2.

Therefore, even integers are integers that are even.

There are 5 consecutive even integers with a sum of –80.

let

x = first integers

Therefore,

x + x + 2 + x + 4 + x + 6 + x + 8 = - 80

5x + 20 = - 80

5x = -80 - 20

5x = - 100

divide both sides by 5

x = - 100 / 5

x = -20

Therefore,

greatest integer = - 20 + 8 = - 12

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Step-by-step explanation:

A. total mass = 9 + (x × 1.2)

B. 21 - 9 = 12kg

12 ÷ 1.2 = 10 articles.

hope it helps. :)

The rewritten expression will be 4a³ and the value of this expression is -500.

Given in the question:

(2a⁴)² / a⁰a⁵

We know that

(xy)ⁿ = xⁿ × yⁿ ... (i)

We can use equation (i) for the next step of our given expression.

In the case of the numerator:

(2a⁴)² = 2² × a⁸ = 4a⁸

Now, we know, any number to the power of zero is 1.

So, for the denominator, we get:

a⁰a⁵ = 1 × a⁵ = a⁵

We know,

pˣ / pʸ = y ⁽ˣ⁻ʸ⁾ ... (ii)

Using this equation, we can further simplify the given expression:

⇒ 4a⁸ / a⁵

⇒ 4 a⁽⁸ ⁻ ⁵⁾

⇒ 4 a ³ ... (iii)

Now, in the second part of the question, we need to find the value of the expression.

Given a = -5

Putting this value in equation (iii) we get:

4 a ³

⇒ 4 (-5) ³

⇒ 4 × (-125)

⇒ -500

Hence, the simplified expression is 4 a ³ and the value of the simplified equation is -500.

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Oliver has to pay an amount of B: $90 with simple interest.

We know simple interest (SI) is given by SI = (p×r×t)/100, where

p = principle, r = rate in percentage, and t = time in years.

Given, Oliver borrowed $1,500 to purchase a riding lawn mower for his lawn care business.

The bank offered him an interest rate of 12% for 6 months.

P = $1500, r= 12% and t = 6 months or 1/2 or 0.5 years.

∴ SI = (1500×12×0.5)/100.

SI = 15×6.

SI = $90.

The amount of simple interest Oliver has to pay is $90.

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∛128 is  the radius r and height h of the cylinder with minimal surface area .

What is Surface Area?

       V = πr²n

   128π = πr²n

        n = 128/r²

s = 2πrn + 2πr²

s = 2πr 128/r² +  2πr²

s = 256π/r + 2πr²

ds/dt = - 256π/r²  + 2πr

d²s/dt² = -256/(-2) r³ + 2π

d²s/dt² = 128/r³ + 2π

for maxima or minima

ds/dt = 0

- 256 π/r² + 2πr = 0

256 = 2r³

 r = ∛128

d²s/dt² = 128/(∛128)³  + 2π

           = 2π + 1

d²s/dt² > 0

so surface area is minimum

         a + r  = ∛128

  n = 128(∛128)²

   n = ∛128

r = ∛128  ,  n= ∛128

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The inverse function is given as s⁻¹ = [tex]\frac{t^2+4t-12}{t^2-9t+14}[/tex] and the equation represents the average sales each day for the real estate company would be [tex]s^{-1}=\frac{t+6}{t-7}[/tex].

What is an equivalent expression?

The equivalent is the expression that is in different forms but is equal to the same value.

A real estate company balances the books for its business on the first day of each month.

It hopes to sell houses every other day of the month.

The average number of houses, S, the company sells each day, t is represented by the inverse of the function given below.

               [tex]s^{-1}=\frac{t^2+4t-12}{t^2-9t+14}[/tex]

Then the inverse function can be written as

                   [tex]s^{-1}=\frac{t^2+4t-12}{t^2-9t+14}\\\\s^{-1}=\frac{t^2+6t-2t-12}{t^2-7t-2t+14}\\\\s^{-1}=\frac{(t+6)(t-2)}{(t-7)(t-2)}\\\\s^{-1}=\frac{t+6}{t-7}[/tex]

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Answer:

C. [tex]S=\frac{7t+6}{t-1}[/tex]

Step-by-step explanation:

I got it right on the exam.

First, take the equation and inverse it, replace and t with y and the y with t

[tex]y=\frac{t^{2}+4t-12}{t^{2}-9t+14} \\\\t=\frac{y^{2} +4y-12}{y^{2}-9y+14}[/tex]

Then, factor both the top and bottom

[tex]t=\frac{(y+6)(y-2)}{(y-7)(y-2)}[/tex]

You see that there is a duplicate of the (y-2) in the numerator and denominator, so they cancel out,

[tex]t=\frac{(y+6)}{(y-7)}[/tex]

Now you can start getting the y alone to make an equation,

[tex]t(y-7)=\frac{(y+6)}{(y-7)} (y-7)\\t(y-7)=(y+6)[/tex]

[tex]ty-7t=y+6\\ty-7t-y+7t=y+6-y+7t\\ty-y=7t-6\\y(t-1)=7t-6\\\frac{y(t-1)}{(t-1)} =\frac{7t-6}{t-1} \\y=\frac{7t-6}{t-1}[/tex]

Answer:

Two negatives, positive answer:

-3 - (-4) = 1

Two negatives, negative answer:

-45 - (-1) = -44

Answer:

y=3+5x

Step-by-step explanation:

Add 5x to both sides.