Use trigonometry to work out the height, v,
of the equilateral triangle below.
Give your answer in millimetres to 1 d.p.
9.5 mm
v

By summing the areas of minor rectangles and triangles, the area of the irregular polygon whose vertices are (- 3, 4), (2, 4), (4, 2), (2, - 3) and (- 3, - 3) is equal to 42 square units.

In this problem we have to determine the area of a irregular polygon, whose calculation is defined as a sum of the areas of triangles and rectangles. To determine the combination of squares and triangles, we need first to plot the vertices and construct the figure on a Cartesian plane.

The area of the irregular figure is:

A = (5) · (7) + 0.5 · 2² + 0.5 · (2) · (5)

A = 35 + 2 + 5

A = 42

By summing the areas of minor rectangles and triangles, the area of the irregular polygon whose vertices are (- 3, 4), (2, 4), (4, 2), (2, - 3) and (- 3, - 3) is equal to 42 square units.

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Answer: B

Step-by-step explanation:

Given:
36=<1.02D-31.2<=41

Solve the compound inequality for D:

36+31.2<=1.02D<=41+31.2

67.2<=1.02D<=72.2

65.9<=D<=70.8

Therefore, option (b) is correct answer

A chord is a part of a circle that joins two points on its circumference but does not pass through the center of the circle. From the given diagram in the question, GH = HJ. Thus the measure of chord HJ is option C. 8.6 units.

A chord is one of the parts of a circle which does not pass through its center but joins two points on its circumference. So that it is not the same as the radius of the circle.

In the given question, it has been stated that: GH and HJ are congruent.

So that by the congruency property of two lines, the length of chod GH is the same as the length of chord HJ.

Therefore, the measure of chord Hj is 8.6 units. This implies option C.

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Answer:

The zeroes are 2, -2, -6, 5.

Step-by-step explanation:

If 2 of the zeroes are 2 and -2 then (x - 2)(x + 2) is factor of the polynomial.

We can now do long division.

Note (x - 2)(x + 2) = x^2 - 4, so we have:

x^2 - 4 ( x^4 + x^3 - 34x^2 - 4x + 120 ( x^2 +   x  - 30 <--- Quotient

             x^4            - 4x^4

                      x^3 -  30x^2 - 4x

                      x^3                -4x

                                  -30x^2   + 120  

                                  -30x^2   + 120

x^2 +   x  - 30 = 0        

(x + 6)(x - 5) = 0

x = -6, 5.              

Answer:

5 and -6

Step-by-step explanation:

So when a polynomial has zeroes at a, then one of the factors (x-a). This is because in factored form, a polynomial can be written using it's factors as such: [tex]a(x\pm b)(x\pm c)(c \pm d)...[/tex] and so on, with the number of factors depending on the degree. The sign is plus or minus because it can be either, it will depend on the sign of the zero. The reason the zeroes are factors, is because when one of the factors is zero it makes the entire thing zero because multiplying them out will always get 0, if you're multiplying by a zero. Anyways, if you have a factor, you can divide, by those factors. So let's do that.

So you have the two factors (x+2)(x-2). So let's divide by each factor. In this example I'll be using long division. So essentially what long division, is finding how many times the leading coefficient of the dividend (x^4+x^3-34x^2-4x+120) can be divided by the leading coefficient of the divisor ((x-2)(x+2)). And then whatever you get, that will be part of the quotient, then take whatever you got, and multiply it by the divisor and then subtract that product from the dividend and then continue this process until the dividend is equal to 0 or you can't divide any further./

Original equation:

[tex](x+2)\div (x^4+x^3-34x^2-4x+120)[/tex]

Divide the leading coefficients:

[tex]\frac{x^4}{x}=x^{4-1} = x^3[/tex]

The x^3 will be part of the quotient, now multiply this by the divisor

[tex]x^3(x+2) = x^{3+1} + 2x^3 = x^4 + 2x^3[/tex]

Subtract the product from the dividend

[tex](x^4+x^3-34x^2-4x+120) - (x^4+2x^3) = x^4+x^3-34x^2-4x+120 - x^4 - 2x^3[/tex]

Group like terms:

[tex](x^4 - x^4) + (x^3 - 2x^3) -34x^2-4x+120[/tex]

Simplify:

[tex]-x^3 -34x^2-4x+120[/tex]

Divide leading coefficients:

[tex]\frac{-x^3}{x} = -x^{3-1} = -x^2[/tex]

The -x^2 will be part of the quotient, now multiply this by the divisor:

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

Subtract this from the dividend:

[tex](-x^3 -34x^2-4x+120) - (-x^3 - 2x^2) = -x^3 -34x^2-4x+120 + x^3 + 2x^2[/tex]

Group like terms:

[tex](-x^3 + x^3) + (-34x^2 + 2x^2)-4x+120[/tex]

Simplify:

[tex]-32x^2 -4x -120[/tex]

Divide the leading coefficients:

[tex]-\frac{32x^2}{x}=-32x^{2-1} = -32x[/tex]

This will be part of the quotient, now multiply this by the divisor:

[tex]-32x(x+2) = -32x^2 - 64x[/tex]

Subtract this from the dividend

[tex](-32x^2 -4x -120) - (-32x^2-64x) = -32x^2 -4x -120 + 32x^2 + 64x[/tex]

Group like terms:

[tex](-32x^2+32x^2) + (-4x+64x) -120[/tex]

Simplify:

[tex]60x+120[/tex]

Divide the leading coefficients:

[tex]\frac{60x}{x} = 60[/tex]

This will be part of the quotient, multiply it by the divisor:

[tex]60(x+2) = 60x+ 120[/tex]

Subtract this from the dividend:

[tex](60x+120)-( 60x+120) = 0\\[/tex]

Since this comes out perfectly to zero there is no remainder, and this is expected since it's a factor. You now take all the quotients from dividing the leading terms and you get [tex]x^3-x^2-32x+60[/tex]. To keep the original equation you simply multiply it, and you get: [tex](x+2)(x^3-x^2-32x+60)[/tex]. But what we really care about is the x^3-x^2-32x+60. We now divide it by the other factor x-2 using the same process

Original equation:

[tex]x^3-x^2-32x+60[/tex]

Divide leading terms:

[tex]\frac{x^3}{x} = x^2[/tex]

This will be part of the quotient, multiply it by the divisor

[tex]x^2(x-2) = x^3-2x^2[/tex]

Subtract this from the dividend

[tex](x^3-x^2-32x+60) - (x^3 - 2x^2) = x^2-32x+60[/tex]

Dividing leading coefficients:

[tex]\frac{x^2}{x} = x[/tex]

This will be part of the quotient, multiply it by the divisor:

[tex]x(x-2) = x^2-2x[/tex]

Subtract this from the dividend:

[tex](x^2-32x+60) - (x^2-2x) = -30x+60[/tex]

Divide the leading terms:

[tex]\frac{-30x}{x} = -30[/tex]

This will be part of the quotient, now multiply it by the divisor:

[tex]-30(x-2) = -30x+60[/tex]

Subtract it from the dividend:

[tex](-30x + 60) - (-30x + 60) = 0[/tex]

Now take all the quotients of the leading terms and you get:

[tex]x^2+x-30[/tex]

Now we can factor this quadratic to get the equation in completely factored form: We're looking for factors that multiply to -30 and add to 1. So we really need to look for factors that have a difference of 1, Or in other words |a-b| = 1. The sign can be determined after we fine the two factors. The factors 5 and 6 have a difference of 1, and since it needs to add to 1, the 5 has to be negative, and the 6 has to be positive.

So we get the factors: [tex](x-5)(x+6)[/tex]. Setting each of them equal to zero gets you:

x-5 = 0

x = 5

x + 6 = 0

x = -6

So the other two zeroes are 5 and -6

Answer:

[tex] \frac{ - 0.5}{ - 0.5} = 1 \\ [/tex]

1 is a whole number, an integer, and rational.

Step-by-step explanation:

Answer:

1)

Let the number of guests = x

∴ Pin hall total charge = 500 + 15x

Bloom place total charge = 350 + 18x

2)

The charges are equal.

∴  500 + 15x = 350 + 18x

⇒ 150 = 3x

⇒ 50 = x

∴ x = 50

This means that the charges for both halls will be equal when there are 50 guests.

The probability that the number of people who show up will exceed the capacity of the plane is 0.002607722

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

The probability that x people show up follows a binomial distribution,

[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}[/tex]

Here, n represents the number of booked people and p represents the probability that a person shows up.

Then, the probability that x people show up is:

[tex]P(x)=\frac{78!}{x!(78-x)!}*0.90^{x}*(1-0.90)^{78-x}[/tex]

So, the probability that the number of people who show up will exceed the capacity of the plane is:

P(x>76) = P(77) + P(78)

Where P(77) and P(78) are calculated as follows:

[tex]P(77)=\frac{78!}{77!(78-77)!}* 0.90^{77}* (1-0.90)^{78-77} \\ =78*0.00029969*0.10\\ =0.0023375[/tex]

[tex]P(78)=\frac{78!}{78!(78-78)!}* 0.90^{78}* (1-0.90)^{78-78} \\ =1*0.00026972*1\\ =0.00026972[/tex]

Finally, P(x>76) is:

[tex]P(x > 76) = 0.0023375 + 0.00026972\\=0.002607722[/tex]

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Answer:

The linear equation for the line with an x - intercept at (-2,0) and y-intercept at (0,-3) is found as [tex]$y=-\frac{3}{2} x-3$[/tex].

Step-by-step explanation:

A condition is given that a line has an x- intercept at (-2,0) and y - intercept at (0,-3).

It is asked to find a linear equation satisfying the given condition.

Step 1 of 2

Determine the slope of the line.

The points of the intercepts of the line are given as (-2,0) and (0,-3). Next, the formula for the slope is given as,

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$[/tex]

Substitute -3&0 for [tex]$y_{2}$[/tex] and [tex]$y_{1}$[/tex] respectively, and 0&-2 for [tex]$x_{2}$[/tex] and [tex]$x_{1}$[/tex] respectively in the above formula. Then simplify to get the slope as follows,

[tex]$$\begin{aligned}m &=\frac{-3-0}{0-(-2)} \\m &=\frac{-3}{2} \\m &=-\frac{3}{2}\end{aligned}$$[/tex]

Step 2 of 2

Write the equation in the slope-intercept form.

The slope-intercept form of a line is given as follows,

y=mx+b

The coordinates at the y- intercept is (0,-3). Now, as the y- coordinate is -3, so b=-3.

So, substitute -3 for b and [tex]$-\frac{3}{2}$[/tex] for m in the equation y=mx+b, and simplify to get the equation as follows,

[tex]$$\begin{aligned}&y=-\frac{3}{2} x+(-3) \\&y=-\frac{3}{2} x-3\end{aligned}$$[/tex]

This is the required linear equation.

The number of combinations which are possible according to the specifications is; 119,700.

It follows from the task content that the 5ember teams to be formed must contain 3 girls and 2 boys.

On this note, it follows that the combinations possible are as follows

20C(3) × 15C2 = 1140 × 105 = 119,700.

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i. 20 people

ii. 3 people

iii. 26 people

We have that;

3x + 5 people likes apples

3x + 5 people likes banana

2x + 15 people likes at least one of the fruits

The total number of people = 55

Let's find the value of x

3x + 5 + 2x + 15 = 55

5x + 20 = 55

5x = 55 - 20

5x = 35

x = 35/5

x = 7

i. People that likes banana only =  3x + 5 = 3 × 5 + 5 = 15 + 5 = 20 people

ii. None of the fruits = 55 - 2x + 15 = 55 - 3x + 5 - 3x + 5 = 55 - 26 +26

= 55 - 52

= 3 people

iii. At most one fruit = 55 - 2x + 15 = 55 - 2(7) + 15 = 55 - 29 = 26 people

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Considering the given exponential function, we have that:

An exponential function is modeled by:

[tex]y = ab^x[/tex].

In which:

When x = 0, y = 1, hence the initial value is of 1. When x changes by 1, y changes by 1/3, hence the base is of 1/3. Exponential functions have no restrictions, hence the domain is of all real values.

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Answer:

On a coordinate plane, an exponential function approaches y = 0 in quadrant 1 and increases into quadrant 2. It goes through (2, one-ninth), (1, one-third), (0, 1), and (negative 1, 3).

Analyze the graph of the exponential decay function.

The initial value is

✔ 1

The base, or rate of change, is

✔ 1/3

The domain is

✔ all real numbers

Step-by-step explanation:

The initial price in this scenario is > sh 276000 and was therefore sold at a loss.

This is a situation in which the selling price of goods or services is lower than the cost price.

We were told that the selling price is sh 276000 which is less than the initial(cost) price hence he ran at a loss during the business transaction.

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Answer: [tex][-3,7][/tex]

Step-by-step explanation:

The range of a function is the set of y-values.

The value of x in the triangle will be 30°

Let the base angles be x. The vertex angle will be:

= 2(x + x) = 4x

Since the sum of the angles in a triangle is 180°, therefore, 4x + x + x = 180°

6x = 180°

x = 180°/6

x = 30°

Therefore, the angles are 30°, 30°, and 120°. This makes up 180°.

Complete question:

In an isosceles triangle, the vertex angle is twice the sum of the base angles. Fund the angles of the triangle.

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Answer:

Step-by-step explanation:

The is a factor of -3 which makes a vertical stretch on the y-axis of the original f(x) equation making a new g(x) equation.

g(x) = a * f(b(x-h)) + k

a - vertical stretch/shrink y-axis

b - horizontal stretch/shirink x-axis

h - horizontal shift right/left

k - vertical shift up/down.

Desmos

The ordered pair which makes both inequalities true is: D. (3, 0).

In Mathematics, an inequality can be used to show the relationship between two (2) or more integers and variables in an equation.

In order to determine ordered pair which makes both inequalities true, we would substitute the points into the inequalities as follows:

At (0, 0), we have:

y > -2x + 3  

0 > -2(0) + 3

0 > 3 (false).

y < x – 2

0 < 0 - 2

0 < -2 (false)

At (0, -1), we have:

y > -2x + 3  

-1 > -2(0) + 3

-1 > 3 (false).

y < x – 2

-1 < 0 - 2

-1 < -2 (false)

At (1, 1), we have:

y > -2x + 3  

1 > -2(1) + 3

1 > -1 (true).

y < x – 2

1 < 1 - 2

1 < -1 (false)

At (3, 0), we have:

y > -2x + 3  

0 > -2(3) + 3

0 > -3 (true).

y < x – 2

0 < 3 - 2

0 < 1 (true).

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Answer:

`1) 2.5cm, 2)1.25inches 3)2.25inches 4)1.25inches

Step-by-step explanation:

1) This is a centimeter rule, one small line represents 0.1cm and 1 long line represent 1cm, the line in between two long lines represent 0.5 cm

As you can see from A to B, is 2 cm + 0.5cm = 2.5cm.

2)This is found on the other side of rulers, inches.

1 small line represent 0.25 inches, line in between 2 long lines is 0.5 inches.

From S to T, is 1 inch + 0.25 inches = 1.25inches

3)Similar to Question 2.

Length of Pencil= 2 inches + 0.25 inches = 2.25 inches

4)This is a more detailed inch rule.

But length of heart shape = 1 inch + 0.25 inches = 1.25 inches

The value of x is 8 mm.

Similar Figures are the figures in which all the sides of one figure are in same ratio with the other side.

Two figures are given which are similar to each other .

When two figures are in ratio a:b then Area will be in ratio a² : b²

Area of figure 1 = 22 sq.mm

Area of figure 2 = 352 sq.mm

A1 :A2 = 22 /352

a² : b² = 1 / 16

a:b = 1/4

Therefore the bigger figure is 4 times the small figure .

The 4mm side will be 16 mm in the larger figure

The difference will be the value of x

x = 24 -16 = 8 mm

Therefore the value of x is 8 mm.

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Answer:

x = 8

Step-by-step explanation:

Similar shapes are dilations of each other using a scale factor.

As the area of both figures has been given, the scale factor that maps the green figure to the yellow figure can be calculated:

[tex]\implies \textsf{Scale factor (area)} = \dfrac{\textsf{area of yellow figure}}{\textsf{area of green figure}} = \sf \dfrac{352}{22} = 16[/tex]

Area is measured in square units. Therefore, to find the scale factor for length, square root the scale factor for area:

[tex]\begin{aligned}\implies \textsf{Scale factor (length)} & = \sf \sqrt{\textsf{Scale factor for area}} \\ & =\sf \sqrt{16}\\ & = \sf 4 \end{aligned}[/tex]

Use the found scale factor for length to find the perimeter of the yellow figure:

[tex]\begin{aligned} \sf \implies \textsf{Perimeter (yellow fig)} & = \sf \textsf{perimeter (green fig)} \times scale\:factor \\ & = \sf 26 \times 4 \\ & = \sf 104\: mm \end{aligned}[/tex]

Find the missing horizontal lengths of the yellow figure by using the corresponding length of the green figure:

⇒ 4 mm × 4 = 16 mm

⇒ 24 mm - 16 mm = 8 mm

Let the height of the yellow figure be y.

⇒ 24 + 8 + 16 + y + x + (y - x) = perimeter

⇒ 48 + 2y = 104

⇒ 2y = 56

⇒ y = 28 mm

Divide the yellow figure into two rectangles (see attachment).

To find x, sum the areas of the rectangles and equate to 352:

⇒ 224 + 16x = total area

⇒ 224 + 16x = 352

⇒ 16x = 128

⇒ x = 8

Answer:

x ≈ 8.0

Step-by-step explanation:

Using the cosine ratio in the right triangle

cos37° =  =  ( multiply both sides by 10 )

10 × cos37° = x , thus

x ≈ 8.0 ( to the nearest tenth )

does this help? :D

Answer:

1200

Step-by-step explanation:

Ok since the oldest was given 2/5, the middle was given 1/2, and the youngest was given the rest, that means the fraction that the youngest was given has to make 2/5 + 1/2 + ? = 1. So we have the equation: [tex]\frac{2}{5} + \frac{1}{2} + x = 1[/tex]. Now to add the fractions simply multiply 2/5 by 2/2 and multiply 1/2 by 5/5 To get the same denominator: [tex]\frac{4}{10} + \frac{5}{10}+x=\frac{9}{10}+x[/tex]. This means that x is 1/10 , because in adding 1/10 you get the whole 10/10. So 120 is one tenth of the entire prize. This means to get the entire prize you simply multiply by 10. This gives you the equation 120 * 10 = 1200

Using an exponential function, we have that:

a) The equation is: [tex]A(t) = 600000(1.058)^{\frac{t}{2}}[/tex].

b) The house will have a value of $1 million during the year of 2037.

c) The expected value on April 1, 2020, is of $621,520.

An increasing exponential function is modeled by:

[tex]A(t) = A(0)(1 + r)^t[/tex]

In which:

In this problem, the parameters are given as follows:

A(0) = 600000, r = 0.058 each 2 years.

Hence the exponential function is:

[tex]A(t) = 600000(1 + 0.058)^{\frac{t}{2}}[/tex]

[tex]A(t) = 600000(1.058)^{\frac{t}{2}}[/tex]

Item b:

It is the year 2019 + t, for which A(t) = 1000000, hence:

[tex]A(t) = 600000(1.058)^{\frac{t}{2}}[/tex]

[tex]1000000 = 600000(1.058)^{0.5t}[/tex]

[tex](1.058)^{0.5t} = 1.6667[/tex]

[tex]\log{(1.058)^{0.5t}} = \log{1.6667}[/tex]

[tex]0.5t\log{1.058} = \log{1.6667}[/tex]

[tex]t = \frac{\log{1.6667}}{0.5\log{1.058}}[/tex]

[tex]t = 18.12[/tex]

The house will have a value of $1 million during the year of 2037.

Item c:

April 1, 2020 is one year and 3 months = 1.25 years after January 1, 2019, hence the value of the home will be given by:

[tex]A(t) = 600000(1.058)^{\frac{t}{2}}[/tex]

[tex]A(1.25) = 600000(1.058)^{\frac{1.25}{2}}[/tex]

A(1.25) = $621,520.

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Answer:

negative y axis

Step-by-step explanation:

6 radians is between 3pi/2 and 2pi radians, so it lies in tje fourth quadrant.

So, a terminal side is the negative y axis.

Answer:

11. ASA

12. SSS

13. SAS

14. SAS

15. Not congruent

16. SAS

Step-by-step explanation:

11.

12, Oppositely congruent

13. indirectly congruent

14. Indirectly congruent

15.

16. Directly congruent

The area of the shaded region = π(4²) - (4√2)² = 50.27 - 32

The area of the shaded region, to one decimal place is: 18.3 m²

The area of the shaded region = area of the circle - area of the square = πr² - s².

Radius (r) of the circle = 1/2(length of diagonal of the square)

Using Pythagorean theorem, we have:

Diagonal = √[(4√2²) + (4√2²)]

Diagonal = √(32 + 32]

Diagonal = 8 m

Radius (r) = 8/2 = 4

s = 4√2

The area of the shaded region = π(4²) - (4√2)² = 50.27 - 32

The area of the shaded region = 18.3 m²

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The special angles pairs that are formed are identified below.

Special angle pairs are formed by a transversal and the parallel lines it cuts across.

Based on the image given, the special angle pairs are given as follows:

Corresponding angles:

∠1 and ∠5

∠4 and ∠8

∠2 and ∠6

∠3 and ∠7

Alternate interior angles:

∠3 and ∠5

∠4 and ∠6

Consecutive interior angles:

∠4 and ∠5

∠3 and ∠6

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[tex]C_{ij}[/tex] is the entry of [tex]C[/tex] in row [tex]i[/tex] and column [tex]j[/tex]. [tex]C[/tex] has 3 rows and 2 columns, so

[tex]C_{11} = \dfrac{|-2\times1+3\times1|}2 = \dfrac{|-2+3|}2 = \dfrac{|-1|}2 = \dfrac12[/tex]

[tex]C_{12} = \dfrac{|-2\times1+3\times2|}2 = \dfrac{|-2+6|}2 = \dfrac{|4|}2 = 2[/tex]

[tex]C_{21} = \dfrac{|-2\times2+3\times1|}2 = \dfrac{|-4+3|}2 = \dfrac{|-1|}2 = \dfrac12[/tex]

[tex]C_{22} = \dfrac{|-2\times2+3\times2|}2 = \dfrac{|-4+6|}2 = \dfrac{|2|}2 = 1[/tex]

[tex]C_{31} = \dfrac{|-2\times3+3\times1|}2 = \dfrac{|-6+3|}2 = \dfrac{|-3|}2 = \dfrac32[/tex]

[tex]C_{32} = \dfrac{|-2\times3+3\times2|}2 = \dfrac{|-6+6|}2 = \dfrac{|0|}2 = 0[/tex]

Then the matrix is

[tex]C = \begin{bmatrix}\dfrac12 & 2 \\\\ \dfrac12 & 1 \\\\ \dfrac32 & 0\end{bmatrix} = \dfrac12 \begin{bmatrix}1&4\\1&2\\3&0\end{bmatrix}[/tex]

The statement that is true is: D. Randomly selected students were used.

Survey can be defined as the data or information that a researcher or research analyst collected from a population so as to reach or draw a conclusion.

Based on the scenario the student were randomly selected which means that the randomly selected students were used to conduct the survey.

Therefore the correct option is D.

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Answer:

Graph of exponential rising up to the right, through the point (0, 2).

Step-by-step explanation:

I think you mean  f(x) = 3^x + 1.

When x = 0, 3^x = 1, so the graph of f(x) = 3^x will rise up to the right and pass through point (0. 1)

The + 1 translates the graph up 1 unit so

The graph of f(x) = 3^x + 1 passes through the point (0, 2).