IfA+B+C = 180°, prove that:
1) cos² A + cos² B-cos² C=1-2sin A sin B sin C​

Daffynition Decoder is a fun and challenging word puzzle that requires both creativity and logical thinking to solve. It is often found in puzzle books, newspapers, and online puzzle websites.

Daffynition Decoder is a word puzzle game that involves decoding phrases or sentences by using a letter-substitution method. The game provides a list of numbered words, where each word is a cryptic definition of a common word or phrase. The solver's goal is to figure out what each word represents and then use the letters from those words to decode a secret message. The game is typically played by filling in a chart or grid, where each row corresponds to a numbered word in the list, and each column represents a letter in the decoded message. The solver uses the numbered words to fill in the corresponding letters in the grid, eventually revealing the hidden message.

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Complete question:

What is project Daffynition Decoder?

She earned a total of $44 from doing chores for her neighbors.

Amount is a numerical quantity that is used to measure, count, or indicate the size, magnitude, or extent of something. It can be used to refer to a variety of things, including money, people, time, temperature, and even abstract concepts. Amounts are often expressed as a numerical value, but may also be expressed as a word or phrase.

Alice earned $44 doing chores for her neighbors. She spent $15 on a book, leaving her with $29. She then spent half of that, or $14.50, on the necklace, leaving her with $14.50. Therefore, she earned a total of $44 from doing chores for her neighbors.

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Answer:

Step-by-step explanation:

7 + 28 + 7 = 42 units^2     see image below

The derivative of given ƒ(x) is:

[tex]$f'(x) = \frac{3x^2 - 6 - 6\log(x)}{x(x^2+5)^2}$[/tex]

Differentiation is a mathematical process used to find the rate at which a function changes with respect to one of its variables. It is the process of finding the derivative of a function, which is a measure of how much the function changes as its input changes. The derivative is used to analyze the behavior of functions, find maximum and minimum values, and solve optimization problems. It is an important concept in calculus and is used in many areas of mathematics, science, and engineering.

We can use the quotient rule to differentiate ƒ(x):

[tex]f(x) &= \frac{2+3\log(x)}{x^2+5} \ \\ \\ \\f'(x) &= \frac{(x^2+5)\frac{d}{dx}(2+3\log(x)) - (2+3\log(x))\frac{d}{dx}(x^2+5)}{(x^2+5)^2} \\ \\\ &\\= \frac{(x^2+5)\left(0+\frac{3}{x}\right) - (2+3\log(x))(2x)}{(x^2+5)^2} \\\ \\ \\ &= \frac{3x^2-6-6\log(x)}{x(x^2+5)^2}[/tex]

Therefore, the derivative of given ƒ(x) is:

[tex]$f'(x) = \frac{3x^2 - 6 - 6\log(x)}{x(x^2+5)^2}$[/tex]

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Answer: $75

Step-by-step explanation:

Steps  he could  use to solve the quadratic equation by completing the square is 5(x² + 4x) = –7

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where x is the variable and a, b, and c are constants. It typically has two solutions, which can be found using the quadratic formula or factoring the equation.

Solution:

The three steps that Sergey could use to solve the quadratic equation by completing the square are:

1.Write the equation in the form ax² + bx + c = 0

2.Add and subtract (b/2)² to complete the square, resulting in an expression of the form a(x + p)^2 + q = 0

3.Solve for x by isolating (x + p)² and taking the square root, resulting in x = (-b ± sqrt(b² - 4ac)) / 2a

Therefore, the three options that correspond to these steps are:

5(x² + 4x + 4) = –7 + 20

x + 2 = Plus or minus StartRoot StartFraction 27 Over 5 EndFraction EndRoot

5(x² + 4x) = –7

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The closest answer is C. between $40.00 and $45.00.

What is the arithmetic operations?

The four basic mathematical operations are Addition, subtraction, multiplication, and division.

Jennifer bought 14.75 gallons of gasoline at a cost of $2.95 a gallon.

The amount she paid for the gasoline is:

14.75 gallons * $2.95/gallon = $43.56

Rounded to the nearest whole dollar, the amount she paid is $44.

Hence, the closest answer is C. between $40.00 and $45.00.

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Therefore, we can estimate that having a 14 ft by 24 ft brick patio installed would cost around $3,033.33.

Cost is a key concept in economics. The price that was paid in return for any products or services is referred to. Cost is a monetary evaluation of the materials, labour, risks, expenses, and time required to purchase goods and services.

We can use proportions to estimate the cost of having a 14 ft by 24 ft brick patio installed based on the given information.

Let's use x to represent the estimated cost of having a 14 ft by 24 ft brick patio installed.

We can set up a proportion:

15/20 = 2275/x

To solve for x, we can cross-multiply and simplify:

15x = 20 x 2275

15x = 45500

x = 3033.33

So the estimated cost of having a 14 ft by 24 ft brick patio installed is approximately $3,033.33.

Therefore, we can estimate that having a 14 ft by 24 ft brick patio installed would cost around $3,033.33.

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In order to resolve the ambiguity and find the correct solution, we need to use additional information such as the Law of Sines or the Law of Cosines.

what is a Triangle ?

Triangle can be defined in which it consists three sides, three angles and sum of three angles is always 180 degrees.

An ambiguous case in geometry refers to a situation where, given certain information about a triangle, there are two possible solutions for one of the unknown angles or sides. This situation arises in trigonometry when we have to solve a triangle using the Law of Sines or the Law of Cosines.

The conditions that create an ambiguous case with two possible solutions are as follows:

Given two sides and an angle opposite one of the sides:

If we are given two sides and an angle opposite one of the sides, there can be two possible solutions for the triangle. For example, if we are given side lengths of 4 and 5, and an angle of 40 degrees opposite the side of length 4, there are two possible triangles that can be formed.

Given two angles and a side opposite one of the angles:

If we are given two angles and a side opposite one of the angles, there can be two possible solutions for the triangle. For example, if we are given angles of 30 degrees and 60 degrees, and a side length of 4 opposite the 30-degree angle, there are two possible triangles that can be formed.

In both of these cases, the ambiguous case arises when the given information is not sufficient to uniquely determine the triangle.

Therefore,  In order to resolve the ambiguity and find the correct solution, we need to use additional information such as the Law of Sines or the Law of Cosines.

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a) The total cost of purchasing the car that is worth $32,000 with the harmonized sales tax (HST) of 13% is $36,160.

b) By leasing the car instead of purchasing it outright, the customer will save $14,760.

c) The lease options for the lessee after returning the car include:

Lease buyout

Extending the lease

Signing a new lease agreement

Buying out the car and then reselling it.

A lease is a financing option for the use of capital assets.

With a lease, the lessee uses the asset for a specific period and the lessor receives periodic payments.

here, we have,

There are operating and finance (capital) leases.  Though, under new accounting standards, all leases are to be classified as finance leases.

Value of the car = $32,000

Harmonized Sales Tax (HST) = 13%

Cost of the car with HST = $36,160 ($32,000 x 1.13)

Lease Arrangement:

Down payment = $1,000

Financing part = $35,160 ($36,160 - $1,000)

Installment payments = 48

Periodic payments = $425

Total lease cost = $21,400 ($1,000 + 48 x $425)

Savings (advantages) from leasing = $14,760 ($36,160 - $21,400)

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Answer:

6 units to the right

Step-by-step explanation:

Z(3,4) → Z'(9,4)

d = √(9-3)² + (4-4)² = √6²+0² = 6   to the right

The translation of point Z of triangle XYZ from (3, 4) to Z′(9, 4) is a move of 6 units to the right.

In Mathematics, particularly geometry, the translation of a point is the process of moving the point a specific number of units either up, down, left or right. In the case of your triangle XYZ, the original position of point Z is at (3, 4) and Z′ is positioned at (9, 4). The y-coordinate hasn't changed; it remains at 4. This tells us that the translation is not up or down. However, if we observe the x-coordinate, you’ll see that it went from 3 to 9. Changing the x-coordinate moves the point horizontally, so we know our translation is left or right. Lastly, the difference between 3 and 9 is 6. We can therefore determine that the point moved 6 units to the right.

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The gas has a capacity of 22 litres at 0 degrees Celsius and increases by 225 litres for each degree above that.

Volume is the amount of space a thing takes up, whereas capacity is the amount of substance it can hold, such as a solid, liquid, or gas. Volume is typically measured in cubic units, whereas capability can be measured in virtually any other unit, such as litres, gallons, pounds, and so on.

According to the problem statement, the volume of the gas and the temperature T in degrees Celsius have a direct relationship. Furthermore, we know that the gas expands by 225 litres for every degree of temperature rise. With this knowledge, we can create the equation shown below:

V = 22 + 225T

where T is indeed the temperature is degrees Celsius and V is the gas's volume in liters.

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Answer:

Step-by-step explanation:

Let's start by finding out what Denniel's weighted average grade is, given the grading system:

Tests: 40% of 70 = 0.4 x 70 = 28

Classwork: ? (we don't know this yet)

Homework: 0 (since he never does homework)

Weighted average grade: 0.4 x 28 + 0.4 x ? + 0.2 x 0 = 0.4 x ?

We know that his overall weighted average grade is 70, so we can set up an equation to solve for his classwork average:

0.4 x ? = 70 - 0.4 x 28 - 0.2 x 0

0.4 x ? = 42

? = 42 / 0.4

? = 105

So Denniel's classwork average is 105.

The value of sin (x/2) is -√(26/27).

The value of cos (x/2) is -√(2/27)

The value of tan (x/2) is 1/12.

We know that sin(x) = -2/27 in Quadrant 3, which means that x lies between 180 degrees and 270 degrees.

a. To find sin(x/2), we can use the half-angle formula:

sin(x/2) = ±√[(1-cos(x))/2]

Since x lies in Quadrant 3, we know that cos(x) is negative. To find cos(x), we can use the Pythagorean identity:

sin²(x) + cos²(x) = 1

(-2/27)² + cos²(x) = 1

cos²(x) = 1 - (-2/27)²

cos(x) = -√(1 - (-2/27)²)

cos(x) = -25/27

Now we can substitute cos(x) into the half-angle formula:

sin(x/2) = ±√[(1-(-25/27))/2]

sin(x/2) = ±√[(27/27 + 25/27)/2]

sin(x/2) = ±√[52/54]

sin(x/2) = ±√(26/27)

Since x is in Quadrant 3, sin(x/2) is negative.

Therefore;

sin(x/2) = -√(26/27)

b. To find cos(x/2), we can use the half-angle formula:

cos(x/2) = ±√[(1+cos(x))/2]

Again, we know that cos(x) is negative, so we can use the value we found earlier:

cos(x/2) = ±√[(1-25/27)/2]

cos(x/2) = ±√[2/27]

Since x is in Quadrant 3, cos(x/2) is also negative.

Therefore;

cos(x/2) = -√(2/27)

c. Finally, to find tan(x/2), we can use the identity:

tan(x/2) = sin(x)/(1+cos(x))

We already know sin(x) and cos(x), so we can substitute them in:

tan(x/2) = (-2/27)/(1-25/27)

tan(x/2) = (-2/27)/(-24/27)

tan(x/2) = 1/12

Therefore,

tan(x/2) = 1/12

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Step-by-step explanation:

closest to all points on the graph

The drawings, class of the polygons, perimeters and areas are presented as follows;

1. The polygon is a kite

Please find attached the drawing of the kite created with MS Excel

The perimeter is about 15.2 units

The area of the kite is 12 square units

3. The polygon is a scalene triangle

Please find attached the drawing of the scalene triangle created with MS Excel

The perimeter of the scalene triangle is about 15.2 units

The area of the triangle is 7 square units

4. The polygon is a rectangle

Please find attached the drawing of the rectangle created with MS Excel

The perimeter of the rectangle is about 14.1 units

Area of the rectangle is 12 square units

A polygon is a geometric figure with three or more straight sides forming a loop

1. Please find attached the drawing of the polygon created with MS Excel

The points on the quadrilateral are; (-2, 3), (3, 1), (-2, -1),  (-3, 1)

The slope of the segment between (-2, 3), and (-3, 1) is (1 - 3)/(-3 - (-2)) = 2

Slope of the segment between (3, 1), (-2, -1) is (-1 - 1)/(-2 - 3) = 0.4

Slope of segment between (-3, 1) and (-2, -1), is (-1 - 1)/(-2 - (-3)) = -2

Slope of the segment between  (-2, 3) and (3, 1) is (1 - 3)/(3 - (-2)) = -2/5 =  -0.4

The length of the sides are;

Distance between (-2, 3) and (-3, 1) = √5

Distance between (-2, -1) and (-3, 1) = √5

Distance between (-2, -1) and (3, 1) = √(29)

Distance between (-2, -3 and (3, 1) = √(29)

3. Please find attached the drawing of the polygon created with MS Excel

The  points on the figure are;

E(-4, 1), F(-2, 3), G(-2, -4)

The length of the segments are;

Distance between (-4, 1), and (-2, 3) = 2·√2

Distance between (-2, -4), and (-2, 3) = 7

Distance between (-2, -4), and (-4, 1)= √(29)

4. Please find attached the drawing of the polygon created with MS Excel

The points on the figure are;

T(1, -2), U(4, 1), V(2, 3), W(-1, 0)

Distance between T(1, -2), and U(4, 1) = 3·√2

Distance between U(4, 1), and V(2, 3) = 2·√2

Distance between V(2, 3) and W(-1, 0) = 3·√2

Distance between T(1, -2) and W(-1, 0) = 2·√2

The slope of the sides are;

Slope between T(1, -2), and U(4, 1) = 1

Slope between U(4, 1), and V(2, 3) = -1

Slope between V(2, 3) and W(-1, 0) = 1

Slope between T(1, -2) and W(-1, 0) = -1

Therefore;

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Answer:18

Step-by-step explanation:

Answer:

Step-by-step explanation:

pls finish question first

Answer:

-1/5(x-6)^2+3

Step-by-step explanation:

+3 makes the graph go up by 3

-6 makes the graph go right 6

we put a negative outside of (x-6) since this goes down and is a -x^2 parabola

no clue about 1/5 just kept putting numbers outside of (x-6) to eventually get the width.

The rate of speed of the jet is given by S = 98 miles per hour

Speed is defined as the rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude

Speed = Distance / Time

Given data ,

Let the rate of speed of the jet be represented as S

Now , the value of S is

The distance traveled by the jet D = 490 miles

The time taken by the jet to travel 490 miles T = 5 hours

So , the speed of the jet is given by

Speed = Distance / Time

On simplifying , we get

The speed of the jet S = 490 / 5

The speed of the jet S = 98 miles per hour

Now , the distance traveled by the jet in 11 hours be D₁

The time taken to travel D₁ distance is 11 hours

where D₁ = Speed x Time

On simplifying , we get

D₁ = 98 x 11

D₁ = 1,078 miles

Hence , the distance traveled at a speed of 98 mph for 11 hours is 1,078 miles

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The graph of both the exponential functions y = 3(11)ˣ  and y = (1/12)ˣ  can be plotted by assuming some arbitrary values of 'x' that correspond to

different values of 'y'.

A function is said to be n exponential function when the independent variable is the exponent.

For example -: aˣ.

The two exponential functions can be graphed by assuming some arbitrary values of 'x'.

In y = 3(11)ˣ.

At x = 1, y = 33.

At x = - 1, y = 3/11.

At x = 2, y = 363.

So, The points are (1, 33), (- 1, 3/11) and (2, 363).

Now we can plot these points and join them by free hand.

Similarly, y = (1/12)ˣ can also be graphed in a similar manner.

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Based on proportional multiplication, the quantity of medication in grams that the patient, who weighs 63 kg needs in 12 hours is 1,224.3 mg.

Proportional multiplication involves the cross-multiplication of two fractional ratios, which are equated to each other.

Proportions involve the equation of two ratios.

The dosage required by a patient in 4 hours with a weight of 9 kg = 58.3 mg

Proportionately, the dosage required by a patient in 12 hours with a weight of 63 kg = 1,224.3 mg (58.3 x 63/9 x 12/4).

Thus, if a patient weighing 9 kg receives 58.3 mg of medicine in 4 hours, proportionately, in 12 hours, a patient weighing 63 kg will receive 1,224.3 mg.

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The future population of the rabbit in the first year is 290.

Logistic growth is defined as the population growth where the growth rate gets smaller as the size of the population approaches the maximum. This maximum point is called carrying capacity.

Here given that,

Carrying capacity, K = 2000

Initial population, P₀ = 200

Growth rate, r = 50% = 0.5

Population in the first year, P₁ = P₀ + r [(1 - P₀/K) P₀]

                                                 = 200 + 0.5 [(1 - 200/2000) 200]

                                                 = 200 + 90

                                                 = 290

Hence the population in the first year is 290.

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The value of  (p o g)(-1) is -2 and the value of (q o p)(-1) is 15 which represents the composite of the function.

It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.

We have a function p and q:

p(x) = 2x - 2

q(x) = x² - 1

= (p o q)(-1)

Find the value of q(-1):

q(-1) = 0

Plug the above value in the function p(x):

(p o q)(-1) = p(q(-1)) = -2

= (q o p)(-1)

Find the value of p(-1):

p(-1) = -4

(q o p)(-1) = q(p(-1)) = 15

Thus, the value of  (p o g)(-1) is -2 and the value of (q o p)(-1) is 15 which represents the composite of the function.

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In linear equation, 3.99 does the amount of money left on the gift card change each month.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

                          Equations whose variables have a power of one are called linear equations. A single variable example with real values for a and b and x as the variable is ax+b = 0.

new difference = [tex]a_{m + 1} - a_{m}[/tex] = answer

            let m = 1

  that's month 1 & 2

         50 - 3.99( m + 1) - (50 - 3.99(m))

         50 - 3.99 - 50

  hence  0 + 0 - 3.99

            = - 3.99

the amount of money left on the gift card change each month by  - 3.99

{that is it reduce by 3.99 }

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The shape οf the graph is similar tο the letter W; hοwever, the regiοn οf interest will lie οn the right οf the οrigin, which means οnly the right half οf the W(a negative number οf cans cannοt be prοduced).

The prοfit prοduced will be maximized if the number οf cans prοduced is greater than 3. The cοmpany will be at a lοss if the number οf cans prοduced is near 2.5. The break even situatiοns are present if the cοmpany prοduces 2 οr 3 cans.

A pοlynοmial functiοn is a functiοn that can be expressed as a sum οf pοwer functiοns in οne variable, where each term has a cοnstant cοefficient and a nοn-negative integer expοnent.Tο sketch the graph οf the functiοn [tex]P(x) = x^4-x^3-11x^2+9x+18[/tex], we can start by analyzing the end behaviοrs οf the functiοn and identifying where the functiοn crοsses the x-axis (i.e., where P(x) = 0).

Tο find the x-intercepts, we need tο sοlve fοr the rοοts οf the equatiοn P(x) = 0. This can be dοne by factοring the expressiοn οr by using numerical methοds. After sοme algebraic manipulatiοn, we can factοr the expressiοn as fοllοws:

[tex]P(x) = (x-3)(x+1)(x^2-8x-6)[/tex]

Therefοre, the rοοts οf the equatiοn P(x) = 0 are x = 3, x = -1, and x = 4 + √(7) οr x = 4 - √(7). The first twο rοοts indicate that the cοmpany breaks even when x = 3 οr x = -1, which means that the cοmpany needs tο prοduce and sell at least 3 οr -1 cans οf sοda tο make a prοfit οf zerο.

With this infοrmatiοn, we can sketch the graph οf the functiοn by plοtting the x-intercepts and the end behaviοrs. The graph will be symmetric abοut the vertical line x = 1.5 (the average οf 3 and -1) and will have a glοbal minimum at the pοint (1.5, -19.5). The graph will alsο have twο additiοnal rοοts at x = 4 + 2√(7) and x = 4 - 2²(7).

In summary, the graph οf the functiοn [tex]P(x) = x^4-x^3-11x^2+9x+18[/tex] has a "U" shape with a glοbal minimum at (1.5, -19.5). The cοmpany will break even when it prοduces and sells at least 3 οr -1 cans οf sοda. By analyzing the end behaviοrs and the x-intercepts, we can sketch the graph withοut using technοlοgy.

Hence, the graph of the function would be given below:

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Deck A the safe walk zone must be painted at the points 13.74 units left and right of the edge of the circle, whereas for Deck B it must be painted at 12.68 units.

Maximum Height refers to the vertical distance between the Reference Level and the highest point of a building's roof surface, including parapet walls, mechanical equipment installed on the roof, authorised Signage, and other permissible projections of any type.

The equation of the given circular space station is:

[tex]x^2 + y^2 = 15^2\\\\x^2 + y^2 = 225[/tex]

Given that, a 6-foot person can walk through the circular station.

Substitute the value of y = 6.

[tex]x^2 + 6^2 = 225\\\\x^2 = 225 - 36\\\\x = \sqrt{189}\\ \\x = \pm 13.74[/tex]

Also, Deck B is 2 feet below deck A thus,

[tex]x^2 + 8^2 = 225\\\\x^2 = 225 - 64\\\\x = \sqrt{161}\\ \\x = \pm 12.68[/tex]

Hence, for Deck A the safe walk zone must be painted at the points 13.74 units left and right of the edge of the circle, whereas for Deck B it must be painted at 12.68 units.

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Answer: Let x be the side length of each square cut out from the corners.

After cutting out the squares, the dimensions of the resulting rectangle will be:

length = 360 - 2x

width = 240 - 2x

The height of the box will be equal to the side length of the squares cut out, which is x.

The volume of the box is given by the formula:

V = length × width × height

Substituting the expressions for length, width, and height, we get:

V = (360 - 2x)(240 - 2x)x

Expanding this expression, we get:

V = x(86400 - 120x + 4x^2)

To find the maximum volume, we need to find the value of x that maximizes the expression for V.

We can do this by finding the critical points of the function V(x) and then determining whether these points correspond to a maximum or a minimum.

Taking the derivative of V(x) with respect to x, we get:

V'(x) = 86400 - 240x + 8x^2

Setting V'(x) = 0 to find the critical points, we get:

8x^2 - 240x + 86400 = 0

Dividing both sides by 8, we get:

x^2 - 30x + 10800 = 0

Using the quadratic formula to solve for x, we get:

x = (30 ± √(30^2 - 4(1)(10800))) / 2

x = (30 ± 210) / 2

We can discard the negative solution, since x represents a length and therefore must be positive. Thus, we get:

x = (30 + 210) / 2

x = 120

Therefore, the side length of the squares cut out from the corners should be 120 mm in order to maximize the volume of the box.

To find the maximum volume, we can substitute x = 120 into the expression for V:

V = x(86400 - 120x + 4x^2)

V = 120(86400 - 120(120) + 4(120)^2)

V ≈ 27648000 mm^3

Therefore, the maximum volume of the box is approximately 27,648,000 mm^3, and this maximum is achieved when the side length of each square cut out from the corners is 120 mm.

Step-by-step explanation:

Answer:

[tex]310 ^\circ F[/tex]

Step-by-step explanation:

The formula relating surface temperature in °F and time in t minutes after removal from the oven is given as

[tex]S(t)= 72 + 238 (0.97^t)\\\\\text{The initial temperature is when t = 0}\\\\\text{At t = 0}\\\\S(0) = 72 + 238(0.97^0)\\\\0,97^0 = 1 \text{ since any number raised to 0 = 1}\\\\\text{Therefore}\\S(0) = 72 + 238 \cdot 1\\= 72 + 238\\\\= 310 ^\circ F[/tex]