find the gradient of the curve y = 3x^4 - 2x^2 + 5x - 2 at the points (0, - 2) and (1,4)

The quotient of the expression (7a³ + ab + 3) / (7a³ + ab - 3) divided by 4ab is (7/4a²) + (1/4b) + (3/4ab).

By dividing each term of the numerator by 4ab:

(7a³ + ab + 3) / (7a³ + ab - 3) ÷ 4ab

= (7a³ / 4ab) + (ab / 4ab) + (3 / 4ab)

Simplifying each term individually:

= (7/4a²) + (1/4b) + (3/4ab)

Thus, the quotient of the expression (7a³ + ab + 3) / (7a³ + ab - 3) divided by 4ab is (7/4a²) + (1/4b) + (3/4ab).

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Answer:

A. Because it would simplify this equation.

If the height of the cylinder is 10, the diameter of the base of the cylinder is 8 units.

A right circular cylinder has two congruent circular bases and a curved surface area. To find the diameter of the base, we need to first find the radius of the base.

Let's begin by using the given information to write an equation. The surface area of a cylinder is given by the formula:

Surface Area = 2πr² + 2πrh

where r is the radius of the base and h is the height of the cylinder.

We are given that the surface area of the cylinder is 112π and the height is 10. Substituting these values, we get:

112π = 2πr² + 2πr(10)

Simplifying the equation by dividing both sides by 2π, we get:

56 = r² + 10r

Now we have a quadratic equation in terms of r. We can solve this equation by using factoring, completing the square, or the quadratic formula. Let's use factoring:

r² + 10r - 56 = 0

(r + 14)(r - 4) = 0

The solutions are r = -14 and r = 4. Since the radius cannot be negative, the radius of the base is 4.

The diameter of the base is twice the radius, so the diameter is 8.

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The legs of the big triangle are 14 units and the legs of the small triangle are 7√2 units

From the question, we have the following parameters that can be used in our computation:

The right triangles

The right triangles are special triangles with acute angles of 45 degrees

This means that the relationship between the legs and the hypotenuse of the triangles is

Hypotenuse = Leg√2

Leg = Hypotenuse/√2

Using the above as a guide, we have the following:

Leg = 14√2/√2

Leg = 14

i.e. the legs of the big triangle are 14 units

Leg = 14/√2

Leg = 7√2

i.e. the legs of the small triangle are 7√2 units

See attachment for the triangles

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Answer:

Step-by-step explanation:

The total interest charges to the Roth Family, after 15 years, given the monthly payment would be $ 118, 000

First, find the total amount paid over the 15 years by the Roth Family to be :

= Number of years x Monthly payment x Number of months per year

= 15 x 1, 850 x 12

= $ 333, 000

The total interest charges that the Roth Family paid would be :

= Total amount paid after 15 years - Total amount borrowed as mortgage

= 333, 000 - 215, 000

= $ 118, 000

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The triangles are similar triangles. Then the value of 'x' will be 27 cm.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

The ratio of the matching sides will remain constant if two triangles are comparable to one another.

From the above theorem, the equation is given as,

(x/15)² = 178.2 / 55

Simplify the equation, then we have

(x/15)² = 3.24

x/15 = 1.8

x = 27 cm

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The value of x for each item is given as follows:

28. x = 5.

29. x = 3.44.

For item 28, we apply the crossing chord theorem, which states that the products of the parts of the chords are equal, hence the value of x is obtained as follows:

16x = 10 x 8

16x = 80

x = 5.

For item 29, we apply the two secant theorem, hence the value of x is obtained as follows:

10(x + 10) = 12(12 + 25)

10x + 100 = 444

10x = 344

x = 3.44.

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Answer:

A. All sides of a rhombus are congruent, so if DK = 8, then KL = 8.

B. The diagonals of a rhombus are perpendicular to each other, so if angle KAL = 2x - 8, then we have:

2x - 8 = 90

2x = 98, so x = 49.

C. All sides of a rhombus are congruent, so if DM = 5y + 2 and DK = 3y + 6, then we have:

5y + 2 = 3y + 6

2y = 4, so y = 2, and KL = 5(2) + 2 = 12.

The measure of Angle 2 is 17 degrees.

The measure of angle 2,  the relationships between the given angles formed by the parallel lines and transversals.

1. The angles formed by lines s, t, and y, clockwise from top left, are:

  - Blank

  - Blank

  - (10x + 5) degrees

  - Blank

  - (4x - 7) degrees

  - Blank

2. The angles formed by s and z are:

  - 65 degrees

  - 1

  - Blank

  - Blank

3. The angles formed by z and t are:

  - 2

  - Blank

  - Blank

  - Blank

From the given information, we can deduce the following:

- Angle 1 is congruent to the angle formed by z and t (corresponding angles).

- Angle (10x + 5) degrees is congruent to angle 65 degrees (alternate interior angles).

- Angle (4x - 7) degrees is congruent to angle 2 (alternate interior angles).

Therefore, we can set up the following equations:

65 = 10x + 5     (Equation 1)

2 = 4x - 7       (Equation 2)

Solving Equation 1:

65 - 5 = 10x

60 = 10x

x = 6

Substituting x = 6 into Equation 2:

2 = 4(6) - 7

2 = 24 - 7

2 = 17

Hence, the measure of angle 2 is 17 degrees.

Therefore, the correct answer is:Angle 2 = 17 degrees.

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4. RSTU is a trapezoid because the opposite sides RS and UT are parallel.

5. RSTU is not an isosceles trapezoid because the diagonals are not congruent.

In order to verify that RSTU is a trapezoid, we would have to determine slope of the pair of opposite sides and check whether at least one pair of opposite sides are parallel;

RU ║ ST

Slope of side RU = Slope of side ST

Slope of RU = (y₂ - y₁)/(x₂ - x₁)

Slope of RU = (1 + 3)/(5 + 3)

Slope of RU = 4/8

Slope of RU = 0.5.

Slope of RS = (y₂ - y₁)/(x₂ - x₁)

Slope of RS = (-9 + 3)/(-4 + 3)

Slope of RS = -6/-1

Slope of RS = 6.

Slope of ST = (y₂ - y₁)/(x₂ - x₁)

Slope of ST = (-2 - 1)/(10 - 5)

Slope of ST = -3/5

Slope of ST = -0.6.

Slope of UT = (y₂ - y₁)/(x₂ - x₁)

Slope of UT = (-2 + 9)/(10 + 4)

Slope of UT = 7/14

Slope of UT = 0.5.

Therefore, RSTU is a trapezoid because the opposite sides RS and UT are parallel.

Question 5.

In order to determine whether RSTU is an isosceles trapezoid, we would have to determine length of the diagonals by using the distance formula and check whether they are congruent;

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance RT = √[(-2 + 3)² + (10 + 3)²]

Distance RT = √170 units.

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance US = √[(5 + 4)² + (1 + 9)²]

Distance US = √181 units.

Therefore, RSTU is not an isosceles trapezoid because the diagonals RT and US are not congruent.

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Answer:

  b. 5.9 mL

Step-by-step explanation:

You want the value that represents the 99th percentile of a normal distribution with a mean of 5 mL and a standard deviation of 0.4 mL.

The inverse of the normal CDF function is used to find the X value for a given area. Here, we want X such that P(x < X) = 0.99 for μ = 5 and σ = 0.4.

The attached calculator output shows that is about X = 5.9305. Rounded to 1 decimal place, the volume is ...

  5.9 mL

<95141404393>

The population of the country will be 796 million approximately in the year 2022.

To find when the population of the country will be 796 million, we can set the equation equal to 796 and solve for t:

[tex]A = 433.4e^{(0.032t)}\\796 = 433.4e^{(0.032t)}[/tex]

Divide both sides by 433.4:

[tex]e^{(0.032t)} = 1.836[/tex]

Take the natural logarithm of both sides:

[tex]ln(e^{(0.032t)}) = ln(1.836)[/tex]

0.032t = 0.605

Divide both sides by 0.032:

t = 18.91

Therefore, the population of the country will be 796 million approximately 19 years after 2003. To find the year, we add 19 to 2003:

2003 + 19 = 2022

Therefore, the population of the country will be 796 million approximately in the year 2022.

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Answer:13.6

Step-by-step explanation:

The equation can be used to determine how many points Patrick earned is ;5 + 6 + y = 14.

What is an equation?

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between.  4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

Given that Billy, Kip, and Patrick earned a combined 14 points during their team's game. Billy earned 5 points and Kip earned 6 points.

We can determine the direct answer by taking Billy and Kip's points and subtracting it from the total combined points.

That will give us 11, which means Patrick earned 3 points.

The equation we can use to represent this is ;5 + 6 + y = 14.

Answer:

-12a-1/4a2

Step-by-step explanation:

The value of x is 14

Fractions are described as expression that shows the part of a whole number, a whole variable. a whole element.

The different types of fractions are listed as;

From the information given, we have that;

1/5 ( x - 10 ) = 4/5

This is written as;

(x -10)/5 = 4/5

cross multiply the values, we have;

5(x-10) = 20

expand the bracket

5x - 50 = 20

Collect like terms

5x = 70

Divide the values

x =14

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Answer:2x+1

Step-by-step explanation:

The value of x, considering the vertical angles in this problem, is given as follows:

x = 11.

Vertical angles are angles that are opposite by the same vertex on crossing segments, hence they share a common vertex, thus being congruent, meaning that they end up having the same angle measure.

The vertical angles for this problem are given as follows:

As the vertical angles are congruent, the value of x is obtained as follows:

2x - 4 = x + 7

x = 11.

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The pattern in the table should be completed as follows;

Number of                      1          3        9       27        81

students with pets

Total number                  3         9        27     81        243

of students

A ratio that compares the number of blue squares to the total number of squares is 3 : 4.

In Mathematics and Geometry, a proportional relationship can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would find the constant of proportionality (k) by using various data points as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 3/1 = 9/3

Constant of proportionality, k = 3.

y = 3x

81 = 3x

x = 27

x = 9/3

x = 3

y = 3(81)

y = 243

For the ratio of blue squares to total squares, we have:

18/24 = 3/4 = 3 : 4

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The calculated value of the element d₁₂ is -8

From the question, we have the following parameters that can be used in our computation:

The matrix D given as

D = [-3 -8 6 -4]

The element d₁₂ represents the element located at the first row and the second column

In this case, the element located at the first row and the second column is -8

Using the above as a guide, we have the following:

d₁₂ = -8

Hence, the value of the element d₁₂ is -8

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Answer:

  y > 2

Step-by-step explanation:

You want the solution to the inequality ...

  -3 > 9 -6y

We can add 6y+3 to both sides. This gives ...

  6y > 12

Dividing by 6 tells us the solution for y:

  y > 2

__

Additional comment

We could have solved by subtracting 9 to get ...

  -12 > -6y

Then dividing by -6 reverses the inequality symbol:

  2 < y

We avoid the issue with negative number multiplication or division by making the y-term positive in our solution process.

<95141404393>

The identification of the polygon that are given in the diagram above would be =

Polygon 1 = Isosceles trapezoid

polygon 2 = parallelogram

polygon 3 = Rectangle

polygon 4 = Rhombus

polygon 5 = Square

polygon 6 = trapezoid

Isosceles trapezoid : This is the type that of polygon in which the the top and base are not equal in length, but they are parallel to each other.

Parallelogram : This is defined as the type of polygon that has two pairs of parallel sides.

Rectangle : This has two pairs of sides that are equal and opposite.

Rhombus: All the sides are equal in length with its opposite angles equal .

Square: All sides are equal in length but and all angles are equal to 90°.

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The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

We have,

To find the probability that an antigen test comes back positive, we need to consider both the true positive rate (probability of a positive test given that the person is infected) and the false positive rate.

Now,

Prevalence of coronavirus in California: 3 out of 1000

False positive rate of the antigen test: 0.05 (5 out of 100)

Let's calculate the probability of a positive test result.

The true positive rate can be calculated as 1 minus the false negative rate (probability of a negative test given that the person is infected):

True positive rate = 1 - 0.2 = 0.8 (or 80 out of 100)

The probability of a positive test result can be calculated using Bayes' theorem:

P(Positive test) = P(Positive test | Infected) x P(Infected) + P(Positive test | Not Infected) x P(Not Infected)

P(Positive test) = (0.8 x 3/1000) + (0.05 x 997/1000)

P(Positive test) = 0.0024 + 0.04985

P(Positive test) = 0.05225

Therefore,

The probability that an antigen test comes back positive is approximately 0.05225, or about 5.225%.

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It would take 21 weeks for the population to surpass 16,000.

The insect population P in a certain area fluctuates with the seasons and is estimated by the function P = 15,000 + 2500 sin(πt/52), where t is given in weeks.

For the population to surpass 16,000, we can set up the following equation:

15,000 + 2500 sin(πt/52) = 16,000

Subtracting 15,000 from both sides, we get:

2500 sin(πt/52) = 1000

Dividing both sides by 2500, we get:

sin(πt/52) = 0.4

We know that sin(πt/52) is positive when t is between 0 and 52 and between 104 and 156 since sine is positive in the first and second quadrants. Therefore, we can write:

πt/52 = sin⁻¹(0.4)

Multiplying both sides by 52/π, we get:

t = (52/π) sin⁻¹(0.4)

Using a calculator, we can evaluate sin⁻¹(0.4) to be approximately 0.4115 radians.

t = (52/π) (0.4115)

t = 21.02

Therefore, it would take 21 weeks for the population to surpass 16,000.

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Answer:

78

Step-by-step explanation:

Let the number of hamburgers sold = x.

Then, the number of cheeseburgers sold = 2x.

Total number of sales: x + 2x = 3x

Total number of sales: 234

3x = 234

x = 78

Answer: 78

Answer:

23p -6k - 42p + 30k - 5p

23p - 42p - 5p + 30k - 6k

23p - 47p + 24k

-24p + 24k

24(- p + k)

24( k - p)

Answer:

The answer is

60p + 24k

Step-by-step explanation:

1: Expand

=23p - 6k + 42p + 30k - 5p

step2: collect like terms

=42p + 23p - 5p - 6k + 30k

step 3: add/subtract them to get your answer

=60p + 24k

your answer☝

Answer: x= 10

Step-by-step explanation:

2x+4=24

2x=24-4

2x=20

x=10

Answer:

Step-by-step explanation:

2x+4=4

subtract 4 from both sides

2x=0

divide by 2 on both sides

x=0

The factored expressions of 9x⁴ - 225y⁸ is (3x² + 15y⁴)(3x² - 15y⁴)

From the question, we have the following parameters that can be used in our computation:

9x⁴ - 225y⁸

²⁴

For the expression, we can apply the difference of two squares

Where

9x⁴ = (3x²)²

225y⁸ = (15y⁴)²

Using the above as a guide, we have the following:

9x⁴ - 225y⁸ = (3x²)² - (15y⁴)²

So, we have

9x⁴ - 225y⁸ = (3x² + 15y⁴)(3x² - 15y⁴)

Hence, the factored expressions is 9x⁴ - 225y⁸ = (3x² + 15y⁴)(3x² - 15y⁴)

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Answer:

108 sq ft

Step-by-step explanation:

The formulas are given in the bottom right.

For the triangles*, the base = 8 ft. The height = 6 feet.

Area = (bh)/2.

Area = (8*6)/2 = (48)/2 = 24 sq ft.

There are two of those identical triangles. 24+24 = 48.

(*Another way to look at this is to say that the base = 8+8 = 16. Area = (bh)/2 = (16*6)/2 = 96/2 = 48. It doesn't matter, mathematically. I can't tell if they have tried to show this as 2 triangles or one.)

For the bottom rectangle, area = length x width = 6*10 = 60 sq ft.

Add them up! 24 + 24 + 60 = 108 sq ft.