I need help with number 1

[tex](2x-1)^{2} = 25 \iff (2x-1)^{2} = 5^{2}\\[/tex]

[tex]x > 0 \implies 2x - 1 = 5 \iff 2x = 5 + 1 \iff 2x = 6 \implies \bf x = 3\\[/tex]

[tex](3y-1)^{2} = 36 \iff (3y-1)^{2} = 6^{2}\\[/tex]

[tex]y > 0 \implies 3y-1 = 6 \iff 3y = 6 + 1 \iff 3y = 7 \implies \bf y = \dfrac{7}{3}\\[/tex]

⇒

[tex]\dfrac{3x - y}{x+y} \cdot \dfrac{1}{7} = \dfrac{3(3) - \frac{7}{3} }{7(3+\frac{7}{3} )} = \dfrac{9 - \frac{7}{3} }{7(\frac{16}{3})} = \dfrac{\frac{27-7}{3} }{7(\frac{16}{3})}=\\[/tex]

[tex]= \dfrac{\frac{20}{3} }{\frac{112}{3}} = \dfrac{20}{3} : \dfrac{112}{3} = \dfrac{20}{3} \cdot \dfrac{3}{112} = \dfrac{20}{112} = \bf \dfrac{5}{28}\\[/tex]

Answer:

y=-3x+7

Step-by-step explanation:

find the slope which is (y1-y2)/(x2-x1). therefore the slope is -3. and then put it in point slope intercept form which is (y-y1)=m(x-x1) = (y-1)=-3(x-2) and simplified the expression is y=-3x+7.

Answer:

The inverse of the function is

[tex]f^{-1} (x)=x^2-3[/tex]

Step-by-step explanation:

Given

[tex]f(x)=\sqrt{x+3}[/tex]

Find the inverse.

Write the function as an equation equal to [tex]y[/tex].

[tex]y=\sqrt{x+3}[/tex]

Interchange the variables.

[tex]x=\sqrt{y+3}[/tex]

Square both sides of the equation.

[tex]x^2=y+3[/tex]

Subtract [tex]3[/tex] from both sides of the equation.

[tex]y=x^2-3[/tex]

Replace [tex]y[/tex] with [tex]f^{-1} (x)[/tex].

[tex]f^{-1} (x)=x^2-3[/tex]

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The proportionality constant (mm to cm) is 10 and the complete table is -

9 cm                 90 mm

12.5 cm             125 mm

50 cm               500 mm

88.49 cm          884.9 mm

A proportional relationship between two variables {x} and {y} means that if {x} increases, then {y} increases with it proportionally.

Given is a proportional relationship between two scales.

{ 1 } -

We can write the proportionality constant (mm to cm) as -

{k} = (20 - 10)/(2 - 1)

{k} = 10

{ 2 } -

We can write the complete table as -

9 cm                  9 x 10 = 90 mm

12.5 cm             12.5 x 10 = 125 mm

50 cm               50 x 10 = 500 mm

88.49 cm          88.49 x 10 = 884.9 mm

Therefore, the proportionality constant (mm to cm) is 10 and the complete table is -

9 cm                 90 mm

12.5 cm             125 mm

50 cm               500 mm

88.49 cm          884.9 mm

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Answer:

Below

Step-by-step explanation:

y = 4x^2

dy/dx = 8x  

   at x = 2 this is    16   <===== not sure what they mean by 'estimate' !

The length of GH on the triangle is 12 units

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

The similar triangles

Assuming that

The smaller triangle with length DF gives the larger triangle with length GH

This means that

GH = 3 * DF

Substitute the known values in the above equation, so, we have the following representation

GH = 3 * 4

So, we have

GH = 12

Hence, the length is 12 units

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Answer:

12 ft

Step-by-step explanation:

keep it simple, just givin' the answer :)

Answer:

Red       100

Blue      150

Green    50

Purple    60

Step-by-step explanation:

The total number of people surveyed
= 20 + 30 + 10 + 12
= 72

The relative area of each pie in the pie chart will be in the ratio of

number of people who prefer a color ÷  total number of people

The central angles of each pie category = above ratio x 360°

These angles should add up to 360°

100 + 150 + 50 + 60 = 360

803.84 square inches is the surface area of a cylinder with an 8-inch radius and 8-inch height.

Any solid shape has a surface area equal to the total of the areas of its faces. For instance, to calculate the surface area of a cuboid, we add the areas of all the rectangles that make it up.

The formula for the surface area of a cylinder is given by:

Surface Area = 2πr² + 2πrh

where r is the radius and h is the height of the cylinder, and π (pi) is a mathematical constant approximately equal to 3.14.

Substituting the given values, we get:

Surface Area = 2 × 3.14 × 8² + 2 × 3.14 × 8 × 8

= 401.92 + 401.92

= 803.84 square inches

the surface area of the cylinder with radius 8 inches and height 8 inches is 803.84 square inches.

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The inequality that can be use to represent the situation is 390 + 6x ≥ 739.

Latanya needs to order some new supplies for the restaurant where she works. The restaurant needs at least 739 glasses. There are currently 390 glasses. Each set on sale contains 6 glasses.

Therefore, the  inequality that can be used to determine x, the minimum number of sets of glasses Latanya should buy can be calculated as follows:

Hence,

390 + 6x ≥ 739

where

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The coordinates of each point after the quadrilateral RSTU is rotated 90 degrees about the origin are R' (2, 1), S' (4, 2), T' = (5, 4), and U' = (3, 3).

In Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).

By applying a rotation of 90° counterclockwise to the vertices of triangle RSTU, the coordinates of the vertices of the image are as follows:

(x, y)                               →            (-y, x)

Ordered pair R  = (1, -2) → Ordered pair R' = (-(-2), 1) = (2, 1).

Ordered pair S  = (2, -4) → Ordered pair S' = (-(-4), 2) = (4, 2).

Ordered pair T  = (4, -5) → Ordered pair T' = (-(-5), 4) = (5, 4).

Ordered pair U  = (3, -3) → Ordered pair U' = (-(-3), 3) = (3, 3).

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Mr Smith looks their children 1 times 1

The highest number of days this institution can continue to accept cats without going over its allowed occupancy is included in the inequality (C) x<42.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way.

The majority of the time, size comparisons between two numbers on the number line are made.

In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other.

Hence, inequality emerges from a lack of balance.

So, what we can do is add 6 in 285 till it reaches 300.

258 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 300

6 is added 7 times, then:

6*7 = 42

Then the correct inequality would be x  < 42.

Therefore, the highest number of days this institution can continue to accept cats without going over its allowed occupancy is included in the inequality (C) x<42.

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One key difference between the two graphs is that [tex]$y=x^2$[/tex] is a function, meaning that for each x value there is only one y value. On the other hand,[tex]$x=y^2$[/tex]is not a function.

The graph of[tex]$y=x^2$[/tex] is a parabola that opens upwards, centered at the origin (0,0). The graph of [tex]$x=y^2$[/tex] is also a parabola, but it opens to the right, also centered at (0,0). Both graphs intersect at the origin, where x=y=0.

Another important similarity between the two graphs is that they are both symmetric with respect to the y=x line,

which is the line of symmetry for both graphs. This means that if we reflect one of the graphs across the line y=x, we get the other graph.

One key difference between the two graphs is that [tex]$y=x^2$[/tex] is a function, meaning that for each x value there is only one y value.

On the other hand,[tex]$x=y^2$[/tex]is not a function, because for some y values there are two corresponding x values (one positive and one negative). This means that the graph of [tex]$x=y^2$[/tex] fails the vertical line test.

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The function is decreasing on the interval -6 < x < 0

increasing on the interval 0 < x < 4, and increasing on 4 < x < 9

In square prism , A cracker box in the shape of a square prism with a base side length of   [tex]SA - \frac{2S^{2} }{4S} = h[/tex].

What is square prism?

The bases of a square prism are square, and the other four faces are rectangles. A square prism is a cuboid object in three dimensions.

                                    Angles and sides that are in opposition to one another are congruent. A magnet bar is an illustration of the shape of a square prism.

we know that'

The formula to calculate the surface area of a square prism is

          [tex]Q = 2s^{2} + 4sh[/tex]

  subtract 2s² both sides

    SA - 2s² = 4sh

  Divide by 4s both sides

     [tex]SA - \frac{2S^{2} }{4S} = h[/tex]

   Rewrite

   [tex]SA - \frac{2S^{2} }{4S} = h[/tex]

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Answer:

See below.

Step-by-step explanation:

We are asked for a unknown number.

To find our number, we should use the Inverse Operation.

The Inverse Operation is the opposite operation.

For example: If we Multiplied, then we should use the Inverse Operation which is Dividing.

Let's divide 256 by -32 to identify our missing number.

Divide:

[tex]\frac{256}{-32} = -8[/tex]

Our missing number is -8.

The range of values for length when the perimeter is between 82 feet and 124 feet is: 27 ≤ l ≤ 48

A closed path that covers, encircles, or outlines a one-dimensional length or a two-dimensional shape is called a perimeter. The area encircling a two-dimensional figure is known as its perimeter. Whether it is a triangle, square, rectangle, or circle, it specifies the length of the shape. The two primary characteristics of a 2D shape are area and perimeter. The perimeter of the shape can be calculated by summing the measurements of its sides and edges. A rectangle's perimeter is equal to twice the product of its length and width.

Given a rectangular fence.

The perimeter P is at least 82 feet and is no more than 124 feet.

82 ≤ P ≤ 124

The width is 14 feet.

The range of values of the length is to be found.

Now,

Perimeter of rectangle = 2( length + width)

The lowest value of the perimeter is 82

82 = 2(l + 14)

82 = 2l + 28

54 = 2l

l = 27

The highest value of perimeter is 124

124 = 2(l + 14)

124 = 2l +28

96 = 2l

l = 48

Therefore the range of values for length when the perimeter is between 82 feet and 124 feet is:

                  27 ≤ l ≤ 48

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The integer number that represents a temperature of 7 ºF below zero is given as follows:

-7 ºF.

Integer number are numbers that can have either positive or negative signal, but are whole numbers, meaning that they have no decimal part.

For temperatures, we have that:

Hence the integer number that represents a temperature of 7 ºF below zero is given as follows:

-7 ºF.

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The factor that both trinomials have in common is; C: (x + 2)

In mathematics, a polynomial is defined an expression that consists of variables that are also referred to as indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x² - 7x - 18

The trinomial of james is;

x² - 7x - 18

Factorizing this gives;

x² + 2x - 9x - 18

= x(x + 2) - 9(x + 2)

= (x + 2)(x - 9)

The trinomial of Kaitlin is;

x² + 14x + 24

Factorizing this gives;

x² + 12x + 2x + 24

= x(x + 12) + 2(x + 12)

= (x + 2)(x + 12)

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The answers to all limits are:

(a) lim x → a [ f(x) + 2g(x)] =  -6,

(b) lim x → a [h(x) − 3g(x) + 1] = 13

(c) lim x → a [ f(x)g(x)] = -8

(d) lim x → a [g (x)]² = 16

(e) lim x → a 3√6 + f(x) / 2g(x)

The central limit theorem states that if you take sufficiently large samples from a population, the samples' means will be normally distributed, even if the population isn't normally distributed.

(a) lim x → a [ f(x) + 2g(x)]

= lim x → a f(x) + 2 lim x → a g(x) (using algebraic limit theorem)

= 2 + 2(-4)

= -6

(b) lim x → a [h(x) − 3g(x) + 1]

= lim x → a h(x) - 3 lim x → a g(x) + lim x → a 1 (using algebraic limit theorem)

= 0 - 3(-4) + 1

= 13

(c) lim x → a [ f(x)g(x)]

= lim x → a f(x) * lim x → a g(x) (using algebraic limit theorem)

= 2 * (-4)

= -8

(d) lim x → a [g(x)]²

= [lim x → a g(x)]^2 (using algebraic limit theorem)

= (-4)^2

= 16

(e) lim x → a 3√6 + f(x) / 2g(x)

= [3√6 + lim x → a f(x)] / [2 lim x → a g(x)] (using algebraic limit theorem)

= [3√6 + 2] / [2(-4)]

= (-3√6 - 2) / 8

Therefore, the answers to all limits are:

(a) lim x → a [ f(x) + 2g(x)] =  -6,

(b) lim x → a [h(x) − 3g(x) + 1] = 13

(c) lim x → a [ f(x)g(x)] = -8

(d) lim x → a [g (x)]² = 16

(e) lim x → a 3√6 + f(x) / 2g(x)

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16 + 5

Because Dan has 16 pencils and he gets 5 more

The values in the table using the expression (time=2 × no. of pages) will be 0.5, 5 and 8.5.

In mathematics, an expression is a combination of numbers, variables, and operations that can be evaluated to produce a value.

Expressions can be simple or complex, and can involve arithmetic, algebraic, or logical operations. Some examples of expressions include:

In each of these examples, the expression consists of one or more terms that are combined using operations such as addition, subtraction, multiplication, division, or exponentiation.

Now,

As given pages = 8 and time= 16 minutes

this can be deduced by expression Time=2 × no. of pages

Then for time= 1,10 and 17 minutes

no. of pages will be 1/2,10/2 and 17/2.

Hence, the table will be:

Pages           8       0.5         5        8.5

Minutes      16        1           10       17

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Answer:

Net profit is the amount of money that a company has earned after deducting all of its expenses, taxes, and other costs from its total revenue. It is the final result of a company's income statement, which shows the revenue, expenses, gains, and losses incurred during a specific period of time, typically a year. Net profit is often used as a key measure of a company's financial performance and is a crucial metric for investors and stakeholders in evaluating the success of a business.

Step-by-step explanation:

Using inequalities, Y(x=0) = -2 and X(y=0) = -2 in region 1.

To pick area 1, draw a line at the intersection of (-2,0) and (0, -2), then choose the right side. such as.

Region 2) To pick the region to the left or upwards in region 2, we must draw a line from (0,2) to (-2,4).

Y(x=0) = 2\sX(y=-2) = -4

Region 3) To pick the region to the right, we must draw a line connecting (-4,0) and (0,4), as I'll demonstrate.

Inequalities in mathematics describe the relationship between two non-equal numbers. Equal does not necessarily mean unequal. When two values are not equal, we typically use the "not equal symbol ()". However, several inequalities are utilised to compare the numbers, whether it is less than or higher than.

The feasible area is now visible. Let's determine the vertices' coordinates.

Where the blue and green lines intersect is our first vertex.

3x + 2 = x + 4 and y = 3x + 2

3x - x = 4 - 2

y = 3× (1) + 2 = 3 + 2 = 5 when 2x = 2x = 1

(X, Y) = Vertex 1 (1,5)

The intersection of the red and green lines is at vertex #2:

y = -x -2; y = x + 4; y = x + 4 = -x - 2; and y = x + x = -2 - 4

2x = -6, x = -6/2, and y = (-3) + 4 = -3 + 4 = 1 are the results.

(X, Y) = Vertex 2 (-3,1)

The intersection of the red and blue lines is at vertex 3:

y = -x - 2, y = 3x + 2 -x - 2, and y = 2 + 2 -4 x = 4 x = -4 / 4 = -1.

So, y = 3× (-1) + 2 = -3 + 2 = -1

(X, Y) = Vertex 3 (-1, -1)

Let's now modify the graph by including the supplied equation.

The intercept where Y=0 is 0 is given by f(x,y) = -3x + 5y = 0 5y = 3x + 0 y = 3/5 × X + 0

the gradient m = 3/5

A line can be drawn from (0,0) to (-10, -6)

Let's now examine how it would appear in the area where it is practical.

We can see that the function intercepts the blue line or the line of the second section, which is where the largest value is located.

y = 3x + 2 3x + 2 (3/5 - 3) x = 2 -12/5 × x = 2 x = -5/6.

Hence, y = 3x + 2 3/5 × (-5/6) = -1/2.

The maximum is therefore found at (x,y) = (-5/6, -1/2).

The point where the function crosses the red line now marks the minimum.

y = 3/5 × x; y = -x -2

3/5 x = -x - 2\s (3/5 + 1) × x = -2\s8/5 x = - 2\sx = -2× (5/8) = -5/4.

y = 3/5 × (-5/4) = -3/4

The minimum is thus found at (x,y) = (-5/4, -3/4)

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According to the preceding statement, then standard deviation becomes 4.19 if the sample variance is given as 17.56.

A measurement of a data set's departure from the mean is referred to as "standard deviation" (or ""). Whereas a large variance shows that the numbers are more spread, a lower number suggests that its data are clustered around the mean. The standard deviation is a measure of how volatile your collected data is generally. It displays the standard deviation for every score with respect to the mean.

We must consider the square root of both the sample covariance matrix to determine the standard deviation.

= √17.56

≈ 4.19

Hence, their standard deviation is around 4.19

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The number of square units for the area of the square is greater than the number of units for the perimeter.

The area of a square is calculated using the formula below:

Area of square = (length of side)²

The length of the side of the square = 10 units

Ares of the square = 10 * 10

Area of the square = 100 units

The perimeter of a square = 4 * length of the side

Perimeter of a square = 4 * 10

The perimeter of a square = 40

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Complete question:

One side of a square is 10 units. Which is greater the number of square units for the area of the square or the number of units for the perimeter?

Answer:

y = 4x + 1

Step-by-step explanation:

Let's graph it in slope intercept form y = mx + b

m = the slope

b = y-intercept

m = 4

b = 1

So, the equation is y = 4x + 1

The profit made from Thembile selling the second hand sewing machine if the cost price is R5700 and the selling price is R8850 is R3150.

Profit/loss refers to the difference between the cost price and selling price of an item. When the difference is positive; it is profit. When the difference is negative; it is loss.

Cost profit = R5700

Selling price = R8850

Profit = Selling price - Cost price

= R8850 - R5700

= R3150

Ultimately, Thembile made a profit of R3150 on the second hand sewing machine.

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