A 3-gallon bottle of fabric softener costs $40.32. What is the price per pint?

Answer:

[tex]y = x \ln 2 +\left(\ln 2 \right)^2-2 \ln 2[/tex]

Step-by-step explanation:

Differentiation is an algebraic process that finds the gradient of a curve.

At a point, the gradient of a curve is the same as the gradient of the tangent line to the curve at that point.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule for Differentiation}\\\\If $y=f(u)$ and $u=g(x)$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4.5 cm}\underline{Differentiating $x^n$}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4.5 cm}\underline{Differentiating $\ln x$}\\\\If $y=\ln x$, then $\dfrac{\text{d}y}{\text{d}x}=\dfrac{1}{x}$\\\end{minipage}}[/tex]

Differentiate the given function using the chain rule:

[tex]\begin{aligned}f(x) & = \left(\ln x\right)^2\\\implies f'(x)& = 2\left(\ln x\right)^{2-1} \cdot \dfrac{\text{d}}{\text{d}x} \ln x\\& = 2\left(\ln x\right)^{1} \cdot \dfrac{1}{x}\\& = \dfrac{2}{x} \ln x \end{aligned}[/tex]

To find the gradient of the function at x = 2, substitute x = 2 into the differentiated function:

[tex]\implies f'(2) = \dfrac{2}{2} \ln 2 = \ln 2[/tex]

Therefore, the gradient of the function at x = 2 is ln(2).

Substitute x = 2 into the function to find the y-value of the point on the curve when x = 2:

[tex]\implies f(2)= \left( \ln 2\right)^2[/tex]

Slope-intercept form of a linear equation:

[tex]y=mx+b[/tex]

where:

Substitute the point (2, (ln 2)²) and the found gradient into the slope-intercept formula and solve for b:

[tex]\begin{aligned} y & = mx+b\\\implies \left(\ln 2 \right)^2 & = \ln 2 \cdot 2 + b\\\left(\ln 2 \right)^2 & =2\ln 2 +b\\b & = \left(\ln 2 \right)^2-2 \ln 2\end{aligned}[/tex]

Therefore, the tangent has the equation:

[tex]y = x \ln 2 +\left(\ln 2 \right)^2-2 \ln 2[/tex]

The circumference is the perimeter of a circle or ellipse in geometry.

The circumference of the ride is about 138.23 feet.

The circumference is the perimeter of a circle or ellipse in geometry. That is, the circumference would be the circle's arc length if it were opened up and straightened out to a line segment. In general, the perimeter is the length of the curve around any closed figure.

The radius is half the diameter.

So, the radius of the circular ride is 22 feet.

Circumference of a circle = [tex]\pi[/tex] d

Where the value of d = 44, then

substitute the value of d in the above equation

Circumference of a circle = [tex]\pi[/tex] d

Circumference of a circle = [tex]\pi[/tex]44

Circumference of a circle = 138.33

Therefore, the circumference of the ride is about 138.23 feet.

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Answer:

he descends only 57 feet ,as the mine is 10 feet below the surface befor descending

Answer:

The coordinate of points A, B, and C will be (0, a), (0, 0), and (a, 0), respectively.

A triangle is a three-sided polygon with three angles. The angles of the triangle add up to 180 degrees.

In an isosceles right triangle, the two legs of the triangle are congruent and intersect at right angles and their opposite angles are also equal.

The vertice C lies on the x-axis, then the coordinate of point C will be (a, 0).

The vertice A lies on the y-axis, then the coordinate of point C will be (0, a).

Then the coordinate of B will be at the origin which is (0, 0).

Thus, the coordinate of points A, B, and C will be (0, a), (0, 0), and (a, 0), respectively.

The diagram is attached below.

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Answer:

Step-by-step explanation:

Determine the two consecutive multiples of 10 that bracket 19.

19 is between 10 and 20.

15 is the midpoint between 10 and 20.

As illustrated on the number line, 19 is greater than the midpoint (15)

Therefore, 19 rounded to the nearest is 20

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The corresponding area and perimeter of the shapes assuming all are unit squares are {5,5) units² and {16,15} units.

A square is a geometrical figure in which we have four sides each side must be equal and the angle between two adjacent sides must be 90 degrees.

Let's assume all blocks that exist in shape are square with dimension units or one.

Now the area of the square is given as ;

A = Side² = 1² = 1

Since the number of squares in 20 is 5

So,

A = 5 unit²

The number of squares in 21 is also 5

So,

A = 5 unit²

Now,

The perimeter of the square as given;

P = 4× side

Since side in 20 are 16 so 16 units.

The side in 21 is 15 so 15 units.

Hence "The corresponding area and perimeter of the shapes assuming all are unit squares are {5,5) units² and {16,15} units".

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Answer: 55

Step-by-step explanation

[tex]\sqrt{a^{2} + b^{2}}=c -- Pythagorean theorem[/tex]

[tex]\sqrt{14^{2} + 20^{2}}=c[/tex] ≈24.41 rounding up to whole number = 24

∠θ = [tex]tan^{-1} (opposite/adjacent) = tan^{-1} (20/14) = 55[/tex]

therefore, ∠θ = 55

Answer: Area= 804.248

Step-by-step explanation:

Using Arithmetic progression(A.P.), the value of a22 will be 107

The formula to calculate A.P. is a + (n-1)d

The value of a5 = 5a1

a1 = 23

a5 = -5a1

a5 = 5 (23)

a5 =  115

a22 = a + (n-1)d

a22 = 23 + (22-1)d

a22 = 23 + 21d                                     ................(1)

a5 = 115

115 = a + (n-1)d

115 = 23 + (4)d                                    ...................(2)

from (1) and (2)

a5 = a + 4d

d = 23

a22 = 107

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Answer:

g(x)=f(x-2)+3

Step-by-step explanation:

The equation which can be used to represent the height of the door in centimeter is 900 = hb

Area of a rectangle = Height × width

900 = h × b

900 = hb

The door's height if its base is 25 centimeters wide?

Area of a rectangle = Height × width

900 = hb

900 = h × 25

900 = 25h

h = 900 /25

h = 36 centimeters

Therefore, the door's height if its base is 25 centimeters wide is 36 centimeters

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After solving each system, 2m = -4n - 4, 3m + 5n = -3, the values of m and n are calculated to be 4 and -3 respectively.

The two given equations, 2m = -4n - 4, and 3m + 5n = -3 are considered to be linear equations since the highest power of the variables in these equations is one.

By rearranging the second equation, the value of m is calculated to be,

3m + 5n = -3

3m = -3 - 5n

m = -3 - 5n / 3

Putting the calculated value of m in the first equation,

2m = -4n - 4

2(-3 - 5n / 3) = -4n - 4

-6 - 10n / 3 = -4n - 4

-6 - 10n = 3(-4n - 4)

-6 - 10n = -12n - 12

-10n + 12n = -12 + 6

2n = -6

n = -6/2

n = -3

To solve for m, put n = -3 in the first equation as follows,

2m = -4n - 4

2m = -4(-3) - 4

2m = 12 - 4

m = 8 / 2

m = 4

Thus the value of m and n in these systems are calculated to be 4 and -3 respectively.

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[tex]r_{y = 2} r_{y - axis}[/tex] is  a glide reflection or composition of transformations that can be used to transform ΔABC to ΔDEF.

A glide translation is what?

When a figure is subjected to one transformation before another is applied?

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Answer:

x = -2

Step-by-step explanation:

2x -5 = 3(4x + 5)  step 1

2x - 5 = 12x + 15  step 2

2x - 12x -5 = 15 Step 3

2x - 12x = 15 +5 simplify

      -10x/-10 = 20/-10 step 4

               x = -2

Using proportions, the coordinates of the point 3/10 of the way from (-3,-8) to (12,6) are: (1.5, -3.8).

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.

We want a point C 3/10 of the way from A(-3,-8) to B(12,6), hence:

C - A = 3/10(B - A).

The x-coordinate of the point is found as follows:

x + 3 = 3/10(12 + 3)

x + 3 = 4.5

x = 1.5.

The y-coordinate of the point is found as follows:

y + 8 = 3/10(6 + 8)

y + 8 = 4.2

y = -3.8.

Hence the coordinates are: (1.5, -3.8).

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Answer: Ratio: 2:3

2 ==> 1st term

Step-by-step explanation:

22.78125/4.5=5.0625

5.0625^(1/(7-3))=

5.0625^(1/4)=1.5 ==> Ratio: 1:1.5 ==> 2:3

4.5 / 1.5^(3-1)=

4.5 / 1.5^2=

4.5 / 2.25 = 2 ==> 1st term

The first term (a) in the geometric progression is approximately 2.

The common ratio (r) is 1.5.

Here, we have to find the common ratio (r) and the first term (a) in the geometric progression, we'll use the formula for the nth term of a geometric sequence:

[tex]u_n = a * r^{(n-1)[/tex]

where:

[tex]u_n[/tex] is the nth term,

a is the first term,

r is the common ratio,

n is the position of the term in the sequence.

Given u3 = 4.5 and u7 = 22.78125:

For n = 3:

[tex]u_3 = a * r^{(3-1)}\\4.5 = a * r^2[/tex]

For n = 7:

[tex]u_7 = a * r^{(7-1)}\\22.78125 = a * r^6[/tex]

Now, we have two equations:

[tex]a * r^2 = 4.5\\a * r^6 = 22.78125[/tex]

Divide equation 2 by equation 1:

[tex](a * r^6) / (a * r^2) = 22.78125 / 4.5[/tex]

Simplify:

[tex]r^4 = 5.0625[/tex]

Now, take the fourth root of both sides to find the value of r:

r = √(5.0625)

r ≈ ± 1.5

Since r is the common ratio, it cannot be negative in a geometric progression.

So, the common ratio (r) is 1.5.

Now, we can find the first term (a) using equation 1:

[tex]a * r^2 = 4.5[/tex]

Substitute the value of r (1.5):

[tex]a * (1.5)^2 = 4.5[/tex]

a * 2.25 = 4.5

Divide by 2.25:

a ≈ 4.5 / 2.25

a ≈ 2

Therefore, the first term (a) in the geometric progression is approximately 2.

The common ratio (r) is 1.5.

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The segments joining the base vertices to the midpoints of the legs of an isosceles triangle will split into two equal triangles so they will be congruent.

If two figures are exactly the same in sense of their length side all things then they will be congruent.

If it is possible to superimpose one geometric figure on the other so that their entire surface coincides, that geometric figure is said to be congruent or to be in the relation of congruence.

ΔABC is isosceles where AB = AC

Now,

Line segment AD is intersecting at the midpoint of BC.

Now,

In ΔABD and ΔACD

AB = AC,AD = AD and BD = DC

Since all sides are equal so by SSS congruent property both will be congruent.

Hence "The segments joining the base vertices to the midpoints of the legs of an isosceles triangle will split into two equal triangles so they will be congruent".

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Answer:

f(x) = |x + 2| + 1

Step-by-step explanation:

The graph of f(x) is translated 2 units to the left and 1 unit up

A horizontal translation of 2 units results in the original function being f(x +2)

A vertical translation of 1 unit happens at f(x) + 1

Horizontal translation of 2 units left of f(x) = |x| is f(x+2) = |x + 2|

Vertical translation of 2 units up : f(x) = |x| + 1

Combined we get g(x) = |x + 2| + 1

The given sequence 2, 5, 9, 14,… is not an Arithmetic sequence.

The given sequence is 2, 5, 9, 14,…

Here, [tex] a_{1}[/tex] = 2, [tex] a_{2}[/tex] = 5 and [tex] a_{3}[/tex] = 9.

A sequence is considered as an Arithmetic sequence if their difference is equal. Represented as -

[tex] a_{2}[/tex] - [tex] a_{1}[/tex] = [tex] a_{3}[/tex] - [tex] a_{2}[/tex] = [tex] a_{4}[/tex] - [tex] a_{3}[/tex] and so on

Now, in the given sequence

⇒[tex] a_{2}[/tex] - [tex] a_{1}[/tex] = 5 - 3

[tex] a_{2}[/tex] - [tex] a_{1}[/tex] = 2

⇒ [tex] a_{3}[/tex] - [tex] a_{2}[/tex] = 9 - 5

[tex] a_{3}[/tex] - [tex] a_{2}[/tex] = 4

⇒ [tex] a_{4}[/tex] - [tex] a_{3}[/tex] = 14 - 9

[tex] a_{4}[/tex] - [tex] a_{3}[/tex] = 5

Since, the difference between the successive terms are not same.

∴ The given sequence 2, 5, 9, 14,… is not an Arithmetic sequence.

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Making C the subject of the formula, we have: C = 5(F - 32)/9.

Given the formula, F = 9/5C + 32, we are asked to solve the equation for C, this means we would make C the subject of the formula.

F = 9/5C + 32

Subtract 32 from both sides

F - 32 = 9/5C + 32 - 32

F - 32 = 9/5C

Multiply both sides by 5/9

5/9(F - 32) = 9/5C(5/9)

5(F - 32)/9 = C

C = 5(F - 32)/9

Therefore, making C the subject of the formula, we have: C = 5(F - 32)/9.

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The expression that can be used to find the perimeter of the triangle is P = 12x^2 - 6 and the perimeter when x = 1.5 is 21 inches.

An expression is a number, or a variable, or a combination of numbers and variables and operation symbols.

Now it is given that,

Pablo folds a straw into a triangle with side lengths of 4x^2−3 inches, 4x^2−2 inches, and 4x^2−1 inches

Suppose the sides of the triangle are,

L = 4x^2−3 inches

B = 4x^2−2 inches

H = 4x^2−1 inches

Now since the perimeter is the sum of all the sides.

Thus, Perimeter of the given triangle is given as,

P = L + B + H

Putting the values we get,

P = (4x^2−3) + (4x^2−2) + (4x^2−1)

Expanding the brackets we get,

P = 4x^2 − 3 + 4x^2 − 2 + 4x^2 − 1

Taking alike terms together,

P = 4x^2 + 4x^2 + 4x^2 - 3 - 2 - 1

Now solving we get,

P = 12x^2 - 6

Thus is the expression for the perimeter of the triangle.

Now the perimeter when x  = 1.5

Put x  = 1.5 in the perimeter expression we get,

P = 12(1.5)^2 - 6

Solving we get,

P = 12*2.25 - 6

Solving we get,

P = 27 - 6

Thus we get,

P = 21 inches

this is the required perimeter when x = 1.5

Thus, the expression that can be used to find the perimeter of the triangle is P = 12x^2 - 6 and the perimeter when x = 1.5 is 21 inches.

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Answer:

12x2 −6; 21 inches

Step-by-step explanation:

The sum of (6.9*10^6) and (5.9*10^5) is 7.49 * 10^6.

The arithmetically adding two or more numbers is called sum.

Scientific notation is a form of presenting a very small or very large number in simple form. it is in the form of multiplication of one digit with 10 raise to the power of respective exponent. plus power is used for huge number and minus power is used for tiny number.

Given numbers = 6.9 * 10^6 and 5.9 * 10^5

5.9 * 10^5 = 0.59 * 10^6

Sum of both numbers

= 6.9*10^6 + 0.59*10^6

= (6.9+0.59)*10^6

= 7.49*10^6

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Answer:

1 25/ 26

2 3/4

3 35/24

6 9/2

8 2/5

Step-by-step explanation:

The measure of JL from the given parameters is 54

The midpoint of a line is the line that divides the line into two equal parts. If K is the midpoint of JL, then the measure of KK is equivalent to the measure of JK. Mathematically;

JK = KL

JK + KL = JL

Given the following parameters

JK=8x + 11 and

KL=14x- 1

Equate

8x+11 = 14x-1

-6x = -12

x = 2

Determine JL

JL = 2(8x+11)

JL = 2(16+11)

JL = 54

Hence the measure of JL from the given parameters is 54

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Answer:  2.5 Hours. I hope I'm not sure but I think you just divide Miles Driven by MPH

Answer:

najua unanijuki bait ni sawa Bora

The formula to calculate the 10th term of a Geometric Progression is :

                         => [tex]a_{10} = a.r^{9}[/tex]          

The formula to calculate [tex]n^{th\\}[/tex] term in a given Geometric progression is

                         =>  [tex]a_{n} = a.r^{n-1}[/tex]  

Where a1 = a, r is a common ratio defined by the ratio of any two successive terms r = a2/a1, and n is the term to be found.

In the given sequence, we have the first term i.e. a = -3, n = 10,  

common ratio  =>  r = a2/a1        =>    r = -2

Calculating using the G.P. formula for the 10th term

                        => [tex]a_{10} = a.r^{9}[/tex]

                        => [tex]a_{10} = (-3).(-2)^{(10-1)}[/tex]

                        => [tex]a_{10\\}[/tex] =  (-3)(-512)  = 1536

The 10th term for this sequence will be 1536.

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Answer:

Perimeter  = 9x+4x+18x+16

Perimeter = 31x+16

Give Brainliest please :)

The first term of the arithmetic sequence so formed is $20 and the common difference is $10.

A series of numbers called an arithmetic progression or arithmetic sequence (AP) has a constant difference between the terms.

Given:

To find: The first term and common difference of the arithmetic progression so formed.

Finding:

As the initial charge charged by the players is $20, it can be taken as the first term of the arithmetic sequence, at 0 hours; a₁ = 20

Now, since the hourly charge is $10, the given arithmetic sequence of the money charged will increase by $10 per hour; thus d = 10

The arithmetic sequence holds true for 5 hours maximum.

Thus the sequence so formed => [tex]a_n = 20 + (n - 1)10[/tex], 1 ≤ n ≤ 5.

Hence, The first term of the arithmetic sequence so formed is $20 and the common difference is $10.

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