Which graph is defined by the function given below?
y=(x-2)(x+5)
-10
10
A.
10
V
-10
OA. Graph A
OB. Graph B
OC. Graph C
OD. Graph D
-10-
B.
10
-10
10
-10
C.
10
-10
10
D.
10
3

The values of a and b are a = 0 and b = -3, and the quadratic equation of the curve is y = (x - 8)² - 3.

What is a quadratic function?

A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

In this case, the vertex is (8, -3).

So we have -

y = a(x - 8)² - 3

To find the value of a, we need one more point on the parabola.

Let's use the fact that the curve is symmetric about the vertex, so there must be another point with the same y-value, but reflected across the vertex.

Since the vertex is at (8, -3), the reflection would be at (8 - 2a, -3).

So we have -

-3 = a(8 - 2a - 8)² - 3

Simplifying -

0 = a(2a)²

0 = 4a³

So either a = 0 or a is undefined (i.e. the parabola is a horizontal line). But we know the curve has a minimum, so it can't be a horizontal line.

Therefore, a must be 0.

Substituting this into our equation, we get -

y = 0(x - 8)² - 3

y = (x - 8)² - 3

Therefore, a = 0 and b = -3, and the equation is y = (x - 8)² - 3.

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The expression [tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}[/tex] is equivalent to 10 for  [tex]x = 10^{(8/15)}[/tex].

In mathematics, an expοnent (alsο called pοwer οr index) is a shοrthand nοtatiοn used tο represent the repeated multiplicatiοn οf a number by itself.

We can simplify the given expressiοn using the prοperties οf expοnents and rοοts.

First, we can simplify the expression inside the first square root as:

√√x1⁰ = [tex]\rm \sqrt{(x^{10})^{(1/4)} }= (x^{10})^{(1/8)} = x^{(10/8)} = x^{(5/4)}[/tex]

Next, we can simplify the expression inside the second square root as:

[tex]\sqrt{(\sqrt{x\³})} = \sqrt{(x^3)^{(1/4) }}= (x^3)^{(1/8)} = x^{(3/8)[/tex]

Therefore, the entire expression becomes:

[tex]\sqrt{(\sqrt{x1\⁰)}} (\sqrt{(\sqrt{x\³})} = x^{(5/4)} * x^{(3/8) }= x^{(15/8)}[/tex]

Now, we need to find the value of x that makes this expression equal to 10.

[tex]x^{(15/8) }= 10[/tex]

Taking the eighth power of both sides:

[tex]x^{15} = 10^8[/tex]

Taking the 15th root of both sides:

[tex]x = (10^8)^{(1/15)}[/tex]

[tex]x = 10^{(8/15)}[/tex]

the equivalent expression is:

[tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}= x^{(5/4) }* x^{(3/8)} = x^{(15/8)} = (10^{(8/15)})^{(15/8)} = 10[/tex]

Therefore, the expression [tex]\sqrt{\sqrt{x1\⁰} }\sqrt{\sqrt{x\³ }}[/tex] is equivalent to 10 for [tex]x = 10^{(8/15)}[/tex].

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Answer:

12m - 25 =335

Step-by-step explanation:

Membership fee per month is m

There are 12months so membership fee for 12 months is 12m

He is given a discount of $25, so we have 12m - 25

The total amount he paid was 335, 12m - 25 =335

Answer:

Step-by-step explanation:

You want to know the parts of the composition of transformations that will map triangle A to A', then to A''. You also want to know what single transformation accomplishes the same thing.

The long side of triangle A points "south". In triangle A', it points "east". To transform a segment pointing south to one pointing east requires a rotation of +90°, or 90° CCW.

The long side of triangle A' lies along the line y=1. In triangle A'', it lies on the line y=-1. The line of reflection is halfway between these values, at ...

  y = (1 +(-1))/2 = 0/2 = 0

The line of reflection is y=0, the x-axis.

We know the transformation involves a reflection. The line of reflection can be found by finding the midpoints of the original and final vertices of the triangle.

The midpoint of the largest acute angle's vertex is ...

  ((2, -3) +(2, -3))/2 = (2, -3)

The midpoint of the right angle vertex is ...

  ((0, -3) +(2, -1))/2 = (1, -2)

The line through these two points is ...

  x +y = -1

The transformation of A to A'' is a reflection in the line x+y=-1.

The required expression represents the value in dollars of the coin n years after Hector found is [tex]$c(n) = 190 \times (1.1)^n$[/tex].

Let's call the percentage increase in value each year "r". We know that after one year, the coin was worth $209, which is 209/190 = 1.1 times what it was worth when Hector found it. So we can set up an equation:

[tex]$190 \times (1 + r)^1 = 209$[/tex]

Simplifying this equation,

[tex]$1 + r = \sqrt[1]{\frac{209}{190}}$[/tex]

1 + r = 1.1

r = 0.1

So the percentage increase in value each year is 10%. To find the value of the coin "n" years after Hector found it, we can use the formula:

[tex]$c(n) = 190 \times (1 + r)^n$[/tex]

Substituting the value we found for "r", we get:

[tex]$c(n) = 190 \times (1.1)^n$[/tex]

So the expression for the value of the coin "n" years after Hector found it is:

[tex]$c(n) = 190 \times (1.1)^n$[/tex]

Thus, required expression represents the value in dollars of the coin n years after Hector found is [tex]$c(n) = 190 \times (1.1)^n$[/tex].

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Answer:

20 × 180° / 22

[(22 - 2) × 180°] / 22

Step-by-step explanation:

20 × 180° / 22

[(22 - 2) × 180°] / 22

Answers:   Choice A and choice E.

Explanation:

The formula to get the sum of all interior angles of a polygon is

S = 180(n-2)

where n is the number of sides and S is the sum of the angles.

We then divide that over n to get the measure of each interior angle. This applies for regular polygons only.

i = interior angle

i = 180(n-2)/n

Now plug in n = 22.

i = 180(n-2)/n

i = 180(22-2)/22

i = 180(20)/22

i = (20*180)/22

which appears to match with choices A and E. This is assuming the comma should be a division symbol.

The following equation can be used to describe isotope decay: [tex]N(t) = N₀e^(kt) (kt)[/tex]where: The quantity of thulium-167 extant at time t is denoted by N(t) (in days).

N0 is the starting concentration of thulium-167 (in grams). decay constant is denoted by k. The natural logarithmic base is e. (approximately 2.71828). We may utilise the stated decay rate of -0.75 to obtain the constant k. k = ln(1 - 0.75) / (-1) (-1) k = 0.2877 As a result, the equation for thulium-167 decay is: N(t) =[tex]N₀e^(0.2877t)[/tex] (0.2877t) We'd want to know how long it takes for 500 g of the isotope to decay toequation can be used to describe 250 g. We'll refer to this as t1. We may The following equation can be used to describe isotope decay.

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Check the picture below.

[tex]\underset{in~degrees}{\textit{sum of all interior angles}}\\\\ S = 180(n-2) ~~ \begin{cases} n=\stackrel{number~of}{sides}\\[-0.5em] \hrulefill\\ n=4 \end{cases}\implies S=180(4-2)\implies S=360 \\\\[-0.35em] ~\dotfill\\\\ (40)+(2x)+( ~~ 360-(x+100) ~~ )+(x)~~ = ~~360 \\\\\\ (40)+(2x)+( ~~ 360-x-100 ~~ )+(x)~~ = ~~360\implies 2x+300=360 \\\\\\ 2x=60\implies x=\cfrac{60}{2}\implies \boxed{x=30}[/tex]

The bottom whiskers cover all data values ​​from the smallest value to Q1, which is the lowest 25% of data values. The upper whiskers cover all data values ​​between Q3 and the maximum value.

The horizontal axis covers all possible data values. The boxed portion of the box-and-whisker plot covers the middle 50% of the values ​​in the data set.

Each whisker covers 25% of the data values.

The lower whisker covers all data values ​​from the minimum value to Q1, i.e. the lowest 25% of data values. The upper whisker

covers all data values ​​between Q3 and the maximum value, i.e. the highest 25% of data values.

The median is inside the box and represents the center of the data. 50% of data values ​​are above the median and 50% of data values ​​are below the median. Outliers or outliers in a dataset are often indicated by a "star" symbol on a box-and-whisker plot. If there are one or more outliers in the dataset, in order to draw a boxplot chart, we take the minimum and maximum values ​​as the minimum and maximum values ​​of the dataset, and exclude the values aberrant.

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The coefficient of 30 means that the mechanic charges $30 per hour for their services, and the $50 in the equation represents the fixed charge.

A linear equation is an algebraic equation of the first degree that forms a straight line when graphed. It has the form y = mx + b, where m is the slope and b is the y-intercept.

Example: y = 3x + 2.

The linear equation y = 30x + 50 represents a mechanic's charges, where y is the total charges and x is the number of hours worked.

The coefficient in this equation is 30, which is the slope of the line. The slope represents the rate of change, which in this case is the hourly rate that the mechanic charges for their services.

Specifically, the coefficient of 30 means that the mechanic charges $30 per hour for their services, and the $50 in the equation represents the fixed charge (or base charge) that's added regardless of the number hours worked.

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The standard normal curve is symmetric about 0.

what is normal curve?

The normal curve, also known as the Gaussian distribution or bell curve, is a type of probability distribution that is commonly used in statistics and probability theory. It is a continuous probability distribution that is characterized by its symmetric bell-shaped curve.

The normal curve is defined by two parameters: the mean (μ) and the standard deviation (σ). The mean represents the center of the curve, while the standard deviation controls the spread or variability of the curve.

To find the area under the standard normal curve that lies to the right of 1.34, we can use the fact that the total area under the curve is 1 and the standard normal curve is symmetric about 0. Therefore, the area under the curve that lies to the left of -1.34 is the same as the area under the curve that lies to the right of 1.34.

Thus, the area under the standard normal curve that lies to the right of 1.34 is also 0.0901227.

The property of the standard normal curve that allowed us to answer part is B. The standard normal curve is symmetric about 0.

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The property of the standard normal curve is -

Option B: The standard normal curve is symmetric about 0.

What is normal curve?

The probability density function for a continuous random variable in a system defines the Normal Distribution or Normal Curve.

It is given that the region under the standard normal curve that lies to the left of,-1.34 has area 0.0901227.

Using the property of the standard normal curve that it is symmetric about 0, we know that the area to the right of 1.34 is equal to the area to the left of -1.34.

So, the area under the standard normal curve that lies to the right of 1.34 is also 0.0901227.

Therefore, the standard normal curve is symmetric about 0.

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Answer:

let's use "x" to represent the son's age.

According to the problem, the father's age is five times the son's age, which means the father's age is 5x.

The sum of their ages is given as 60, so we can write an equation:

x + 5x = 60

Simplifying the equation, we get:

6x = 60

Dividing both sides by 6, we get:

x = 10

So the son's age is 10 years old.

To find the father's age, we can use the equation we set up earlier:

father's age = 5x = 5(10) = 50

Therefore, the father is 50 years old.

Answer:

Step-by-step explanation:

Let x = the son’s age

So 5x = the father’s age

60 = x + 5x

60 = 6x

10 = x

Therefore , the solution of the given problem of equation comes out to be perpendicular lines have an ambiguous gradient.

Complex algorithms frequently employ a variable term to guarantee agreement with the two opposing claims. Equations, which are mathematical statements, are used to demonstrate the equality of a number of academic figures. Instead of expression dividing 12 into two parts in this situation, the normalise technique provides b + 6 to use the data from y + 6 instead.

Here,

All locations along a horizontal line have the same y-coordinate and are parallel to the x-axis. Y = c,

where c is a constant, is its expression. A straight line has no gradient because there is no slope to it.

All locations on a vertical line have the same x-coordinate because the line is parallel to the y-axis. x = c, where c is a constant, is its solution. Because a straight line has no slope, its gradient is ill-defined.

In conclusion, the formulae for horizontal and vertical lines are straightforward and easy to recognise, and

their gradients exhibit special characteristics. horizontal lines have "0", however perpendicular lines have an ambiguous gradient.

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Each of the expressions can be rewritten using one of the properties of rational expressions as follows:

1. 49^1/2 * 3/5 = 3.21 (Power of  a power property)

2. 4^3/4 ÷ 4^1/4 = 4^3/4 -1/4 = 2.00 (Quotient of power property)

3. 27^4/3 ÷ 125^4/3 = 0.13 (Power of a quotient property)

4. 16^7/4 + 1/2 = 16^9/4 = 512 (Product of power property)

5. 36^3/2 + 64^3/2 = 216 + 512 = 728 (Power of product property)

6. 9^5/2 ÷ 25^5/2 = 243/3123 =0.78 (Power of a quotient property)

7. 81^5/6 - 1/3 = 81^1/2 = 9 (Quotient of powers property)

8. 64^4/3 ÷ 123^4/3 = 255.65/610.72 = 0.42 (Power of a quotient property)

9. x^13/10 y^2/5 * y^6/5  (Power of product property)

10. 4a^1/2 ÷ 2(4a^3/4) = 4a^-1/4 ÷ 2 = 2a^-1/4 (Quotient of powers property)

11. 27 * 2.99 x^6/5 - 1/5 = 80.91x (Quotient of powers property)

12. x^23/12 x^29/12 y^2     (Power of product property)

A rational expression is a mathematical representation that aims at indicating the ratios of two different or similar polynomial values. The alphabets x and y are often used in such expressions.

It is also possible to rewrite rational expressions using some laid down principles. The given expressions have been rewritten with the relevant rational rules.

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The total number of times you would expect the number 7 to appear is:1000 × 1/6 = approximately 143 times.

If you were to roll two dice 1000 times, the number of times you would expect the number 7 to appear is approximately 143 times.The possible outcomes when two dice are rolled are 6 × 6 = 36. The probability of obtaining a total of 7 when two dice are rolled is:6/36, which reduces to 1/6.

The total number of times you would expect the number 7 to appear is:1000 × 1/6 = approximately 143 times.

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Answer:

Step-by-step explanation:

D SHOULD BE THE ANSWER.

Experimental probability of tails = Number of tails / Total number of tosses

Experimental probability of tails = 287 / 500

Experimental probability of tails = 0.574 or 57.4%

Since the experimental probability of getting tails is greater than the theoretical probability, this suggests that the coin may be biased toward tails.

Answer:

y=1.5x-2

Step-by-step explanation:

Use equation y=mx+c

m is the gradient of the graph

c is the y-intercept

Finding the gradient:

Coordinates of the blue circles: (0,-2) and (2,-5)

(-5)-(-2)=-3

2-0=2

3÷2=1.5=m

Therefore, the gradient is 1.5

Next, to find the y-intercept, just simply look at the y axis

The blue circle is the intercept.

Therefore, the final equation is:

y=1.5x-2

Hope this helped

The increase in the percentage of students at the college who would be on academic probation would be 67% over the current figure.

The anticipated increase of students on academic probation from 3% to 5% of the student body would mean that the increase in the percentage of students at the college who would be on academic probation would be 67% over the current figure.

As given in the question, the administrators of the local college wanted to increase the GPA to remain in good standing from 2.0 and reported that the anticipated increase of students on probation would be from 3% to 5% of the student body. Now, the relative increase in percentage can be calculated as:

Relative increase = (new value - old value) / old value × 100%. Using this formula, we can calculate the percentage increase in students on academic probation at the college as follows: Relative increase = (5 - 3) / 3 × 100%Relative increase = 2 / 3 × 100% Relative increase = 0.67 × 100% Relative increase = 67%

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By answering the presented question, we may conclude that As a result, cylinder the amount of vinegar required to completely fill a cylindrical container with a radius of 5 cm and a height of 10 cm is 824.7 grammes.

The cylinder, which is frequently a three-dimensional solid, is one of the most fundamental curved geometric shapes. In elementary geometry, it is known as a prism with a circle as its basis. A cylinder is also defined as an infinitely curved surface in several modern domains of geometry and topology. A "cylinder" is a three-dimensional object composed of curved surfaces with circular tops and bottoms. A cylinder is a three-dimensional solid figure with two bases that are both identical circles linked by a curving surface at the height of the cylinder, which is defined by the distance between the bases from the Centre. Cold beverage cans and toilet paper wicks are two examples of cylinders.  

V = r2h, where r is the radius and h is the height, gives the volume of a cylinder.

Because the radius is 5 cm and the height is 10 cm in this situation, the volume of the cylinder is:

V = π(5 cm) (5 cm)

²(10 cm) = 785.4 cm³

Because the vinegar entirely fills the cylinder, its volume equals the cylinder's volume. We may calculate the mass of vinegar using its density:

density Equals mass divided by volume

density x volume Equals mass

1.05 g/cm3 x 785.4 cm3 = mass

mass = 824.7 g

As a result, the amount of vinegar required to completely fill a cylindrical container with a radius of 5 cm and a height of 10 cm is 824.7 grammes.

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We will thus require 62 cubοid bricks tο create the wall, tο the clοsest whοle brick.

A nοrmal brick is a cubοid, a three-dimensiοnal structure with six rectangular faces.

We need tο have at minimum 42/18 = 2.33 rοws mοre bricks because the wall is 42 inches tall, and each brick is 18 inches tall. We require three rοws οf bricks since we can't have a half rοw.

We require 144/7 = 20.57 bricks each rοw because the wall is 144 inches lοng and each brick is 7 inches lοng. There must be 21 bricks each rοw because we cannοt have half bricks.

The first rοw requires 21 whοle bricks. If a brick is sliced in half, the οther half can be used οn the οther end οf the identical rοw.

21 whοle bricks are alsο required fοr the secοnd rοw. Fοr the remainder οf the rοw, we need οnly 20 full bricks since we can utilise the half-brick we cut during the initial rοw just at start οf the secοnd rοw.

The third rοw requires 21 whοle bricks. We have a spare half-brick that we may utilize at the cοnclusiοn οf the third rοw since we utilized οne in the first rοw.

Hence, 62 cοmplete bricks are needed, which equals 21 + 20 + 21.

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The square pizza has an area of about 42.06 square inches more than the circular pizza, meaning that the square pizza is approximately 28% larger than the circular pizza.

The area of a circle with a diameter of 14 inches is (π/4) x (14 inches)², which is approximately 153.94 square inches.

The area of a square with sides measuring 14 inches is 14 inches x 14 inches, which is 196 square inches.

Therefore, the circular pizza has an area that is approximately 42.06 square inches smaller than the square pizza. In other words, the square pizza is approximately 28% larger than the circular pizza.

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The factorization of 18y - 21 = 3(6y - 7)

Factorization is the simplification of an expression by grouping similar term together into smaller terms.

Since we have the expression 18y - 21, and we desire to factorize it, we proceeds as folows.

Since we have 18y - 21. we write each term out as a product of their common factors.

So,

18y - 21 = 3 × 6 × y - 3 × 7

Factorizing out the common factor 3, we have that

18y - 21 = 3 × 6 × y - 3 × 7

18y - 21 = 3(6 × y - 7)

= 3(6y - 7)

So, we have the expression as 3(6y - 7)

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The equation of the line perpendicular to PQ passing through Z is:

y - ZY = 21/29(x - ZX)

To solve this problem, we can use the fact that the centroid of a triangle divides each median into two segments, where the length of the segment adjacent to the vertex is twice the length of the other segment.

Let M be the midpoint of WY, N be the midpoint of WX, and Z be the centroid of triangle WXY. Then, we know that:

WM = MY

WN = NX

ZM = 2ZC

ZN = 2ZB

where ZB and ZC are the lengths of the segments of the median from vertex W that intersect XY at points B and C, respectively.

We can use this information to set up a system of equations:

WM + WN = WY

ZM + ZN = ZW

ZC + ZB = ZX

Substituting the given values, we have:

(87/2) + (39/2) = WY

2ZC + 2ZB = WZ

48 + (87/2) = ZX

Simplifying each equation, we get:

WY = 63

ZC + ZB = WZ/2

ZX = 141

Now, we can use the fact that the sum of the lengths of the medians of a triangle is equal to three times the length of the segment connecting the centroid to the circumcenter. That is:

WX + WY + WZ = 3 × ZO

where O is the circumcenter of triangle WXY.

Substituting the given values and the values we found earlier, we have:

WX + 63 + 141 = 3 * ZO

Simplifying, we get:

WX = 3ZO - 204

To solve for WX, we need to find ZO. We know that the circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle. Let P be the intersection of the perpendicular bisectors of WY and WX, and Q be the intersection of the perpendicular bisectors of WY and XY. Then, ZO is the distance from Z to the line PQ.

We can find the equation of PQ by finding the midpoint of WY and WX, which is M. Then, the slope of PQ is the negative reciprocal of the slope of the line passing through M and Z.

We know that the slope of the line passing through M and Z is:

(ZM - MY) / (ZW - WM) = (2ZC - MY) / (WZ - WM)

Substituting the given values and the values we found earlier, we have:

(2ZC - 63/2) / (141/2 - 87/2) = (2ZC - 63/2) / 27

Solving for ZC, we get:

ZC = 117/2

Now, we can find the equation of PQ. The midpoint of WY is (0, 63/2), and the midpoint of WX is (87/2, 0). Therefore, the slope of PQ is:

(-87/2) / (63/2) = -29/21

The equation of PQ is therefore:

y - 63/2 = (-29/21)(x - 87/2)

Simplifying, we get:

29x + 21y = 1911

To find ZO, we need to find the distance from Z to the line PQ. We know that the equation of the line perpendicular to PQ passing through Z is:

y - ZY = 21/29(x - ZX)

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Complete Question

If Z is the centroid of ΔWXY, WR=87, SY=39, and YT=48, find each measure.

a)WS=

b) WY =

c) WZ =

d) ZR =

e) ZT =

f) YZ =

Answer:

[tex]4\frac{1}{2}[/tex]

Step-by-step explanation:

To write this fraction in its simplest form, we simply have to divide any values in the denominator and numerator that can be simplified!

We can see that 9x and 12x can be simplified by dividing them by 3x!

We are now left with: [tex]\frac{3 + 15}{4}[/tex]

Now we have to add 3 and 15 to simplify further...

We are left with: [tex]\frac{18}{4}[/tex]

The final step we must take is that we have to change this fraction into a mixed fraction...

Hope this helps, have a lovely day! :)

Cost of materials for each boy's costume = 5 * 4 * 4 + 5 * 3 * $.50 = 80 + 7.5 = $87.50 Cost of materials for each girl's costume = 6 * 5 * 4 + 6 * 2 * $2 = 120 + 24 = $144

Let A = [5; 6], B = [4; 5], C = [3; 2], D = [4; 2], E = [$.50; $2].

Cost of materials for each costume = A * B * D + A * C * E

Cost of materials for each costume = [5; 6] * [4; 5] * [4; 2] + [5; 6] * [3; 2] * [$.50; $2]

Cost of materials for each costume = [80; 120] + [15; 24]

Cost of materials for each costume = [95; 144]

Total cost of materials for each costume = $239

Cost of materials for each boy's costume = 5 * 4 * 4 + 5 * 3 * $.50 = 80 + 7.5 = $87.50

Cost of materials for each girl's costume = 6 * 5 * 4 + 6 * 2 * $2 = 120 + 24 = $144

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The exponential function of the line that goes through points (0, 20) and (2, 180) is y = 20 × 3ˣ.

To find the exponential function of a line that goes through two points, we need to use the general form of the exponential function:

y = a × bˣ

where a is the initial value, b is the base, and x is the independent variable (usually time).

We can use the two points given to find the values of a and b.

Let's start with the point (0, 20):

y = a × bˣ

Plug in x = 0 and y = 20

20 = a ×  b⁰

20 = a ×  1

a = 20

Next, let's use the point (2, 180):

y = a × bˣ

Plug in x = 2 and y = 180

180 = 20 × b²

b² = 180/20

b² = 9

b = 3

Now, plug these values into y = a × bˣ

Therefore, the exponential function y = 20 × 3ˣ

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The scale factοr frοm shape A tο shape B is 8

Shapes in variοus dimensiοns can be scaled using the Scale Factοr. In geοmetry, we study many geοmetrical fοrms that exist in bοth twο- and three-dimensiοns. The scale factοr is a way tο cοmpare figures with similar appearances but differing scales οr measurements. Cοnsider twο circles that resemble οne anοther but may have different radii.

The scale factοr οf a shape refers tο the amοunt by which it is increased οr shrunk. It is applied when a 2D shape, such as a circle, triangle, square, οr rectangle, needs tο be made larger.

K is the scaling factοr fοr x in the equatiοn y = Kx. This expressiοn can alsο be visualized in terms οf prοpοrtiοnality:

y ∝ x

K can therefοre be regarded as a prοpοrtiοnality cοnstant in this cοntext.

The fundamental Prοpοrtiοnality Theοrem alsο aids in a better understanding οf the scale factοr.

a) We must cοmpare the cοrrespοnding sides οf bοth shapes in οrder tο determine the scaling factοr frοm shape A tο shape B.

It is evident that the sides AB and BC οf shape A and shape B, respectively, cοrrespοnd tο the sides PQ and QR, respectively, οf shape B.

As a result, the scale factοr between shape A and shape B is:

AB/PQ = BC/QR

We can substitute the values given in the diagram:

4/t = 6/(t+4)

Crοss-multiplying:

4(t+4) = 6t

Simplifying:

4t + 16 = 6t

2t = 16

t = 8

b) The fact that the cοrrespοnding sides οf identical fοrms are prοpοrtiοnate can be used tο determine the value οf t.

It is evident that the sides AB and BC οf shape A and shape B, respectively, cοrrespοnd tο the sides PQ and QR, respectively, οf shape B.

As a result, we may cοnstruct the equatiοn shοwn belοw:

AB/PQ = BC/QR

We can substitute the values given in the diagram:

4/t = 6/(t+4)

Crοss-multiplying:

4(t+4) = 6t

Simplifying:

4t + 16 = 6t

2t = 16

t = 8

Therefοre, the value οf t is 8.

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The domain of function f is (-∞, 10) U (10, ∞) or {x|x ≠ 10}.

The range of function f is (-∞, -3) U (-3, ∞) or {y|y ≠ 3}.

The inverse of this function is f⁻¹ = (10x - 4)/(3 + x).

The domain of function f⁻¹ is (-∞, -3) U (-3, ∞) or {x|x ≠ 3}.

The range of function f⁻¹ is (-∞, 10) U (10, ∞) or {y|y ≠ 10}.

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In order to determine the inverse of any function, you should swap both the input value (x-value) and output value (y-value) as follows;

f(x) = y = (3x + 4)/(10 - x)

x = (3y + 4)/(10 - y)

10x - yx = 3y + 4

3y + yx = 10x - 4

y(3 + x) = 10x - 4

f⁻¹ = (10x - 4)/(3 + x)

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a.sin(3θ)cos(θ) - cos(3θ)sin(θ) = sin(4θ) - sin(2θ) = sin(4θ - 2θ) = sin(2θ)

b.2θ = 0, π, 2π, 3π,...

To solve this equation, first use addition or subtraction to simplify. In this case, it is best to add the two terms together and use the trigonometric identity sin(A)cos(B) + cos(A)sin(B) = sin(A+B). Applying this, we can rewrite the equation as:

sin(3θ)cos(θ) - cos(3θ)sin(θ) = sin(4θ) - sin(2θ) = sin(4θ - 2θ) = sin(2θ)

Now, to find all solutions of the simplified equation, use the fact that sin(2θ) = 0 when 2θ = 0, π, 2π, 3π,... Using this, the solutions of the equation are:

2θ = 0, π, 2π, 3π,...

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