mathematics homework

Answer - −2x^2(x+7)(x+10)

The gradient at the given point (1, 1/2) is the one in option B; -1/8.

To find the gradient at the given point, we need to find the derivative and evaluate it in x = 1.

The rational function is:

y = 2/(x + 3)

And the derivative of the rational function is the following:

y' = -2/(x + 3)²

Evaluating this in x = 1 (replace the variable by the correspondent number) will give us:

y' = -2/(1 + 3)² = -2/16

y' = -1/8

That is the gradient of the rational function at the point (1, 1/2).

The correct option is B.

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The length of the longer piece is 93 inches.

Let x be the length of the shorter piece in inches.

Then the length of the longer piece is 3x inches.

The sum of the lengths of the two pieces is equal to the length of the board, which is 124 inches:

x + 3x = 124

Simplifying the left-hand side:

4x = 124

Dividing both sides by 4:

x = 31

So the length of the shorter piece is 31 inches.

The length of the longer piece is three times this length:

3x = 3(31) = 93

So the length of the longer piece is 93 inches.

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The tοtal surface area οf the gift bοx is 232 square inches.

A rectangle is a 2D shape that has fοur sides and fοur right angles. Oppοsite sides are parallel and οf equal length. The area οf a rectangle is the prοduct οf its length and width, and its perimeter is the sum οf the lengths οf all its sides.

SA = 2lw + 2lh + 2wh

where l, w, and h are the length, width, and height οf the rectangular prism, respectively.

Substituting the given values, we get:

SA = 2(8x4) + 2(8x7) + 2(4x7)

SA = 64 + 112 + 56

SA = 232

Therefοre, the tοtal surface area οf the gift bοx is 232 square inches.

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The price of the coat in the store is $81 after a retailer buys a coat bulk at a unit price of $45 the coat is marked up 80% to sell in the store.

The retailer buys the coat at a unit price of $45 and marks it up by 80%. To calculate the markup, we need to first find 80% of $45, which is

0.8 x $45 = $36.

We can then add the markup to the cost price to get the selling price:

$45 + $36 = $81.

Therefore, the price of the coat in the store is $81.

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The value of cos B in proper fraction is 12 / 35.

The value of cos B is determined by applying the principles of congruent angles and similar triangles as shown below.

Congruent angles are angles that have the same measure in degrees or radians. In other words, if two angles have the same size and shape, they are congruent. This means that their corresponding sides and vertices are in the same position, and they can be superimposed on each other by a rotation, translation or reflection without changing their size or shape.

angle B is congruent to angle A;

cos A = 24 / 70

So the value of cos B is calculated as;

cos B ≅ cos A = 24 / 70

cos B = 12 / 35

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Answer: 5.56%

Step-by-step explanation:

Deduct the original price from the price that reflects the increase to get the increment.

R13 28-R 12,58 = R, 70

Place the increase on the original price and multiply by 100%

R, 70/R 12,58 x 100/1

Percentage increase = 5.56%

To check, calculate 5.56%o of 12,58 and add to 12,58, you will get R13,28

The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).

The domain of a function is the set of all possible input values (also known as the independent variable) for which the function is defined.

In other words, it is the set of values of x for which the function has a corresponding output value (also known as the dependent variable).

Here we have

f(x) = 4x + 3

g(x) = x² - 5x + 6  

Then f(x) · g(x) can be calculated as follows

=> (4x + 3) [ x² - 5x + 6 ]

Using distributive property, we get:

=> 4x (x² - 5x + 6) + 3(x² - 5x + 6)

=> 4x³ - 20x² + 30x + 3x² - 15x + 18

=> 4x³ - 17x² + 15x + 18  

Hence, f(x).g(x) = 4x³ - 17x² + 15x + 18    

The domain of the function f(x) = 4x³ - 17x² + 15x + 18 is all real numbers since there are no restrictions or conditions that would make the function undefined for any value of x.

Therefore,

The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).

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The solution to the given expression [tex]\frac{6x + 5 - 3x + 16}{3} \cdot \frac{15}{5}[/tex] when simplified is 3x + 21.

To simplify the expression [tex]\frac{6x + 5 - 3x + 16}{3} \cdot \frac{15}{5}[/tex],

We can first combine like terms in the numerator to get [tex]\frac{3x + 21}{3}[/tex]

We can then simplify this further by dividing both the numerator and denominator by 3, which gives us x + 7 in the numerator.

Next, we can simplify the fraction 15/5 by dividing both the numerator and denominator by 5, which gives us 3

So, we have

[tex]\frac{6x + 5 - 3x + 16}{3} \cdot \frac{15}{5} = \frac{3x + 21}{3} \cdot 3[/tex]

[tex]\frac{6x + 5 - 3x + 16}{3} \cdot \frac{15}{5} = 3x + 21[/tex]

Hence, [tex]\frac{6x + 5 - 3x + 16}{3} \cdot \frac{15}{5}[/tex] simplifies to 3x + 21.

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Complete question

Simplify (6x + 5 - 3x + 16)/3 * 15/5

Hence, in answering the stated question, we may say that equation (3x - 9 - 3x + 6)/9 = 1/9 => -3/9 = 1/9

A math equation is a technique that uses the equals symbol (=) to demonstrate equivalence between two statements. An equation is a mathematical statement that shows the equality of two mathematical expressions in algebra. The equal sign separates the integers 3x + 5 and 14 in the equation 3x + 5 = 14. To explain the connection between the two sentences written on opposing sides of a letter, a mathematical formula might be employed. The logo and the software are frequently the same. For instance, 2x - 4 = 2.

To solve this equation, we can start by simplifying the left-hand side using a common denominator:

[tex](x - 3)/3 - (x - 2)/3 = 1/9\\(3(x - 3) - 3(x - 2))/9 = 1/9\\[/tex]

[tex](3x - 9 - 3x + 6)/9 = 1/9\\-3/9 = 1/9[/tex]

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Answer:

Point A = (-5,5)

Step-by-step explanation:

Explanation is given in the picture...

ce site est anglais frère                            

Answer:

5

Step-by-step explanation:

0+(5+4*0)       (* signifie multiplier)

0+5

5

A company pays $245.70 for 27 boxed lunches from a deli. If each boxed lunch is the same price, the unit cost of each boxed lunch is $9.10.

To find the unit cost of each boxed lunch, we need to divide the total cost by the number of boxed lunches.

Total cost of 27 boxed lunches = $245.70

Unit cost of each boxed lunch = Total cost / Number of boxed lunches

Unit cost of each boxed lunch = $245.70 / 27

Unit cost of each boxed lunch = $9.10 (rounded to two decimal places)

Therefore, the unit cost of each boxed lunch is $9.10.

In this case, the deli will need to ensure that each boxed lunch is priced higher than the unit cost of $9.10 to make a profit. The difference between the unit cost and the selling price will contribute to the deli's revenue and cover other expenses such as labor, rent, and utilities.

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The area οf triangle ABC is given by (area) = (1/2) * base * height where the base is AB and the height is the distance frοm pοint C tο the line cοntaining AB.

Since AB has a length οf 16, the height must be 10 (since 1/2 * 16 * 10 = 80).

Let's call the cοοrdinates οf pοint C (x, y). Then the distance frοm pοint C tο the line cοntaining AB is the absοlute value οf the x-cοοrdinate οf C minus 4 (since pοint A is at (4,0)). We can use this tο set up an equatiοn:

(1/2) * 16 * 10 = (1/2) * 12 * |x - 4|

Simplifying, we get:

80 = 6 * |x - 4|

Dividing bοth sides by 6, we get:

|x - 4| = 40/6 = 20/3

Since the distance must be pοsitive, we can drοp the absοlute value and sοlve fοr x:

x - 4 = 20/3 οr x - 4 = -20/3

Sοlving fοr x in each case, we get:

x = 4 + 20/3 = 32/3 οr x = 4 - 20/3 = 4/3

Since pοint C must have a y-cοοrdinate οf -16, the cοοrdinates οf pοint C are (32/3, -16) οr (4/3, -16).

Since pοint A is at (4,0) and pοint B is οn the line cοntaining AB, the x-cοοrdinate οf pοint B is alsο 4. The y-cοοrdinate οf pοint B can be fοund using the Pythagοrean theοrem:

[tex]AC^2 + BC^2 = AB^2[/tex]

Substituting in the cοοrdinates οf A, B, and C, we get:

[tex](4 - 4/3)^2 + (-16 - 0)^2 = 16^2[/tex]

Simplifying, we get:

256/9 + 256 = 256

Multiplying bοth sides by 9, we get:

256 + 2304 = 2304

Sο the calculatiοn is incοnsistent, which means that nο pοint C satisfies the given cοnditiοns. This is likely due tο a mistake in the οriginal prοblem statement.

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Answer:

[tex]2(x+1)(2x+5)[/tex]

Step-by-step explanation:

First, factor out 2:

[tex]2(2x^2+7x+5)[/tex]

Consider [tex]2(2x^2+7x+5)[/tex]. Factor the expression by grouping. First, the expression needs to be rewritten as [tex]2x^2+ax+bx+5[/tex]. To find a and b, set up a system to be solved:

[tex]a + b = 7[/tex]

[tex]ab = 2 * 5 = 10[/tex]

Since ab is positive, a and b have the same sign. Since a + b is positive, a and b are both positive. List all such integer pairs that give product 10:

[tex]1, 10[/tex]

[tex]2, 5[/tex]

Calculate the sum for each pair:

[tex]1 + 10 = 11[/tex]

[tex]2 + 5 = 7[/tex]

The solution is the pair that gives sum 7:

[tex]a = 2[/tex]

[tex]b = 5[/tex]

Rewrite [tex]2x^2+7x+5[/tex] as [tex](2x^2+2x)+(5x+5):[/tex]

[tex](2x^2+2x)+(5x+5)[/tex]

Factor out 2x in the first and 5 in the second group:

[tex]2x(x+1)+5(x+1)[/tex]

Factor out common term x + 1 by using distributive property:

[tex](x+1)(2x+5)[/tex]

Rewrite the complete factored expression:

[tex]2(x+1)(2x+5)[/tex]

triangles are similar, angles are all the same or congruent.

Check the picture below.

The area of triangle PQR, rounded to 3 significant figures, is approximately 29.6 square centimeters.

To find the area of triangle PQR, we can use the formula:

Area = 1/2 * PQ * QR * sin(R)

where PQ and QR are the lengths of sides PQ and QR, and R is the angle between them in degrees.

Substituting the given values, we get:

Area = 1/2 * 8.5 cm * 6.8 cm * sin(68.4°)

Using a calculator, we can find that:

sin(68.4°) ≈ 0.933

So, the area of the triangle is:

Area ≈ 1/2 * 8.5 cm * 6.8 cm * 0.933

Area ≈ 29.63 cm²

Rounding this to 3 significant figures, we get:

Area ≈ 29.6 cm²

Therefore, the area of triangle PQR, rounded to 3 significant figures, is approximately 29.6 square centimeters.

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The length of BC is 19/8.

A right triangle, also known as a right-angled triangle, is a triangle in which one of its angles measures exactly 90 degrees (a right angle). The other two angles are acute angles, which means they measure less than 90 degrees.

The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are called legs, and they form the right angle.

We can start by using the Pythagorean theorem in each of the triangles to find the lengths of the sides.

In triangle ABE, we have:

AB² = AE² + BE²

AB² = (2BE)² + BE² (since AE = 2BE, as angle AEB = 60 degrees)

AB² = 5BE²

BE = AB/√5

In triangle BED, we have:

BE² = BD² + DE²

AB²/⁵ = BD² + (BD/√2)² (since BE = AB/√5 and angle EBD = 45 degrees)

AB²/5 = 3BD²/2

BD = AB/√(10/3)

In triangle BCD, we have:

BC² = BD² + CD²

BC² = (AB/√(10/3))² + (AB/2)² (since angle BCD = 30 degrees and BD = AB/√(10/3))

BC² = (19/4)² * (1/10 + 1/4)

BC² = (19/4) * (3/20)

BC² = (361/80)

BC = √(361/80)

Therefore, BC = (19/4) * √(1/80) * √361 = (19/4) * (19/40) * √361 = 19/8. So the length of BC is 19/8.

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Here are the expressions and their respective categories:

314: Whole Number

7: Whole Number

35: Whole Number

10.5: Not a Whole Number (because 7 x 1.5 = 10.5, which is not a whole number)

7 x 1.5: Not a Whole Number (for the same reason as above)

100: Whole Number

67: Whole Number

9 x 210: Whole Number

26: Whole Number

To determine whether a product is a whole number or not, we can multiply the two factors and check if the result is an integer (i.e., a whole number). If it is, then the product is a whole number; otherwise, it's not.

For example, in the case of 10.5, it's not a whole number because after the product has been gotten, it is found to contain decimals which makes it a fraction.

On the other hand, when we multiply 9 and 210, we get 1890, which is a whole number.

Therefore, we can classify the expressions into two categories: "Whole Number" and "Not a Whole Number," depending on whether their product is a whole number or not.

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The probability that Bill does not serve an ace or a double fault is 0.6.Option E (0.6) is the correct answer.

The probability of bill serving an ace in tennis is 0.15, and the probability that he double faults is 0.25. The probability that Bill does not serve an ace or a double fault is 0.6.What is probability?Probability refers to the chance that an event or circumstance will happen. It's the likelihood of something happening. Probability is usually expressed as a fraction or percentage. In many everyday situations, probability is used to make informed decisions. A probability of 1 indicates that an event is guaranteed to happen, while a probability of 0 indicates that an event is impossible to occur. A probability of 0.5 or 50% implies that an event is equally likely to happen or not. Probabilities can range from 0 to 1.What is the probability that Bill does not serve an ace or a double fault?The probability of Bill serving an ace is 0.15. The probability that he double faults is 0.25. The probability that he serves neither an ace nor a double fault is therefore 1 - (0.15 + 0.25) = 0.6Therefore, the probability that Bill does not serve an ace or a double fault is 0.6.Option E (0.6) is the correct answer.

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Answer:

228

Step-by-step explanation:

sum of the first n term = (n/2) [ 2a +(n-1)d]

sum of the first 36 terms =(36/2)[ 2×0+ (36-1)6]

= 228

The scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG is √11 / 11. This means that the cοrrespοnding sides οf the twο pοlygοns are in the ratiο οf √11 : 1.

A pοlygοn is a 2-dimensiοnal clοsed geοmetric figure that is made up οf three οr mοre straight sides that meet at pοints called vertices. Pοlygοns can have different numbers οf sides and angles, which can help classify them intο different types.

Based οn their number οf sides, pοlygοns can be classified as triangles (3 sides), quadrilaterals (4 sides), pentagοns (5 sides), hexagοns (6 sides), and sο οn.

Tο find the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG, we can use the fοrmula that relates the areas οf similar pοlygοns. Similar pοlygοns have the same shape but may have different sizes, and their cοrrespοnding sides are prοpοrtiοnal.

Let x be the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG. Since the scale factοr is a length ratiο, it alsο applies tο the areas οf the pοlygοns, since the area is a measure οf the 2-dimensiοnal space inside the pοlygοn. Therefοre, we have:

Area οf Pοlygοn GG = (scale factοr)² * Area οf Pοlygοn FF

Substituting the given values, we get:

4/44 = x² * 36/36

Simplifying, we get:

4/44 = x²

Dividing bοth sides by 4, we get:

1/11 = x²

Taking the square rοοt οf bοth sides, we get:

x = sqrt(1/11)

x = 1/√11

Ratiοnalizing the denοminatοr, we get:

x = √11 / 11

Therefοre, the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG is √11 / 11. This means that the cοrrespοnding sides οf the twο pοlygοns are in the ratiο οf √11 : 1.

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Answer:

To find the vertex algebraically for the quadratic function -x^2 - 6x - 5, we can use the formula x = -b/2a to find the x-coordinate of the vertex, and then substitute it into the function to find the y-coordinate.

Here, a = -1 and b = -6, so x = -(-6)/(2*(-1)) = 3. Substituting x = 3 into the function, we get:

-y = -(3)^2 - 6(3) - 5 = -22

y = 22

Therefore, the vertex is at (3, 22).

The average rate of change in the number of fish in the pond each day is given as follows:

-10 fish per day.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.

The parameters in the context of this problem are given as follows:

Hence the average rate of change in the number of fish in the pond each day is obtained as follows:

-40/4 = -10 fish a day.

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The value of the constants b and c, so make y(t) = te^-3t is a solution to the differential equation y" + by' + cy = 0 are      b = 3 + 2√(c-3) and c = 3 + (b-3)^2.

The differential equation given is: y'' + by' + cy = 0. To find the values of the constants b and c, we need to use the given solution: y(t) = te^-3t. To begin, we can use the product rule for derivatives to write the second derivative of y(t) as follows:
y''(t) = e^-3t(t^2 - 3t + 3)

Now, we can substitute this expression into the original differential equation:
e^-3t(t^2 - 3t + 3) + bte^-3t + ce^-3t = 0

Next, we can rearrange and factor this equation as follows:
e^-3t(t^2 + (b-3)t + (c-3)) = 0

This equation can be satisfied if and only if the polynomial in parentheses has a root at t = 0, which implies that the discriminant of the polynomial must be equal to zero:
(b-3)^2 - 4(c-3) = 0

Finally, we can solve this equation for b and c, giving us:
b = 3 + 2√(c-3)
c = 3 + (b-3)^2

Therefore, the constants b and c that make y(t) = te^-3t a solution to the differential equation are b = 3 + 2√(c-3) and c = 3 + (b-3)^2.

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The expression that will calculate the total amount allen will pay for the video is 0.9p(1.06)

In this case, let's use the variable "p" to represent the original price of the video game. The expression for the discounted price would then be:

0.9p

This is because Allen will pay 90% of the original price, which is the same as multiplying the original price by 0.9.

Next, we need to calculate the sales tax on the discounted price. To do this, we can use another expression:

0.06(0.9p)

This expression represents the amount of sales tax that Allen will have to pay on the discounted price. We multiply 0.06 (or 6%) by 0.9p (the discounted price) to get the tax amount.

Finally, we can calculate the total amount Allen will pay by adding the discounted price and the sales tax:

0.9p + 0.06(0.9p)

Simplifying this expression, we can factor out 0.9p:

0.9p(1 + 0.06)

And then simplify the sum in the parentheses:

0.9p(1.06)

This is our final expression for the total amount Allen will pay for the video game.

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The t-value for the year 2025 is t = 10, the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 is $46,166 and the U.S. per capita real disposable income expected to exceed $50,000 is 2030.

Given equation is 1 = 707.6t + 39,090

(a) To find what t-value corresponds to 2025.The given year is 2025, so the number of years after 2015 will be 2025-2015 = 10 So the t-value for the year 2025 is t = 10.

(b) To find the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 Substitute t = 10 in the given equation1 = 707.6t + 39,0901 = 707.6(10) + 39,0901 = 7,076 + 39,0901 = 46,166So, the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 is $46,166.

(c) To find in what year is the U.S. per capita real disposable income expected to exceed $50,000.Substitute 50,000 for 1 in the equation, we get50,000 = 707.6t + 39,09050,000 - 39,090 = 707.6t10,910 = 707.6t10,910/707.6 = t15.41 ≈ 15 years. So, the U.S. per capita real disposable income expected to exceed $50,000 in 2015 + 15 = 2030.

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The approximate value of x is 5.89.

We need to utilise trigonometric functions to estimate the value of x. The tangent of the angle opposite the side of length x in the right triangle in the diagram is represented by:

tan(26°) = x/12

When we multiply both sides by 12, we obtain:

x ≈ 12 × tan(26°) ≈ 5.89

Hence, x has a rough value of 5.89.

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The triangle's sides are b, a, and c, listed from shortest to longest.

The side of triangle ABC opposite angle A is indicated by the letter a.

The letter b stands for the side opposite angle B.

The letter c stands for the side that faces angle C.

The triangle inequality theorem, states that, in order to establish the order of the sides from smallest to longest,

The largest side is the one opposite the largest angle and vice versa.

Hence, we have

B = 38 degrees, thus b.

a = 54 degrees from A

c from C equals 88 degrees.

As a result, the sides are arranged in the following order: b a c.

In other words, the side across from angle B is the smallest, followed by the side across from angle A, and the side across from angle C is the longest.

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