If the table of the function contains exactiy two potential turning points, one with an input value of -1,which statement best describes all possible values of m

Answer:

9) x = 12 cm

10) x = 10.4 ft (Rounded)

11) y = 39.7 in (Rounded)

Step-by-step explanation:

Just use Pythagorean theorem

Answer and Explanation:

We can solve for x in these questions using the Pythagorean Theorem:

a² + b² = c²,

where a and b are the legs (shorter sides) of a right triangle, and c is its hypotenuse (longest side).

Applying this theorem to the problems at hand:

9. We are using (10 / 2) as one of the legs.

(10 / 2)² + x² = 13²

↓ simplifying division

5² + x² = 13²

↓ subtracting 5² from both sides

x² = 13² - 5²

↓ simplifying the right side

x² = 169 - 25

x² = 144

↓ taking the square root of both sides

x = 12

10.

6² + x² = 12²

↓ subtracting 6² from both sides

x² = 12² - 6²

↓ simplifying the right side

x² = 144 - 36

x² = 108

↓ taking the square root of both sides

x = [tex]\sqrt{108}[/tex]

↓ simplifying the square root

    [tex]\sqrt{108} = \sqrt{3 \cdot 3 \cdot 3 \cdot 2 \cdot 2} = (3\cdot 2)\sqrt{3} = 6\sqrt3[/tex]

x = 6[tex]\sqrt3[/tex]

11.

24² + 33² = y²

↓ simplifying the right side

576 + 1089 = y²

1665 = y²

↓ taking the square root of both sides

y = [tex]\sqrt{1665}[/tex]

↓ simplifying the square root

    [tex]\sqrt{1665} = \sqrt{3 \cdot 3 \cdot 5 \cdot 37} = 3\sqrt{5 \cdot 37} = 3\sqrt{185}[/tex]

y = 3[tex]\sqrt{185}[/tex]

The distance between the two cruise ships after 2 hours is approximately 4.201 miles.

How to find distance?

To solve this problem, we need to use trigonometry to find the distance between the two cruise ships after 2 hours.

In the diagram, A and B are the two cruise ships, and D is the distance between them after 2 hours. The angle between their paths is 35 degrees.

We can use the formula: distance = speed x time

to find how far each cruise ship travels in 2 hours. For Cruise A, the distance it travels is

[tex]distance_A = 18 \times 2 = 36 \: miles[/tex]

For Cruise B, the distance it travels is

[tex]distance_B = 15 \times 2 = 30 \: miles

[/tex]

Now we can use trigonometry to find the distance between the two cruise ships. We can use the tangent function, since we know the angle and the opposite and adjacent sides of the triangle formed by the two cruise ships and the distance between them:

tan(35°) = D / (36 - 30)

Simplifying this equation, we getting,

D = (36 - 30) x tan(35°)

D = 6 x 0.7002

D ≈ 4.201 miles

Therefore, the distance between the two cruise ships after 2 hours is approximately 4.201 miles.

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Answer: I am pretty sure 7

Step-by-step explanation: I think this because 20 is the most and 13 is the leased so 20 -13 =7 but i just got done like a month or 2 ago so i just learned this so dont quote me sorry if it isn't right

Answer:

Step-by-step explanation:

The rule that relates the number of months (m) to the cost (c) of a gym membership is:

c = 15m + 25

To find the cost of a 1 year (12 months) membership, we substitute m = 12 into the equation:

c = 15(12) + 25 = 205

Therefore, the cost of a 1 year membership is $205 (which includes the $25 sign up fee).

Answer:

[tex](x + 18)^{2} + (y + 9)^{2} = 1[/tex]

Step-by-step explanation:

The description table or graph or equation with the correct type of function:

Linear function table: A

Non-linear function graph: B

Linear function equation: C

Non-linear function equation: D

Linear function graph: E

Non-linear function table: F

A function is a mathematical concept that describes a relationship between two sets of values, where each input value (independent variable) corresponds to exactly one output value (dependent variable). In other words, a function is a rule or a formula that takes an input value and produces a unique output value.

Here,

The table of values shows a constant rate of change, which indicates a linear function. We can verify this by calculating the slope between any two points, which should be the same. For example, the slope between (-2, -8) and (2, 0) is (0 - (-8))/(2 - (-2)) = 8/4 = 2. So the function is of the form y = mx + b, where m = 2 and b = -4. This gives us the equation y = 2x - 4, which is a linear function. Therefore, the correct match is A.

The graph shows a curve, which indicates a non-linear function. It does not resemble a straight line, so we can eliminate linear functions. The shape of the graph suggests that it might be a quadratic function, which has a U-shaped curve. Quadratic functions can be represented by an equation of the form y = ax² + bx + c. We can't determine the equation of the function from the graph alone, but we can conclude that it is a non-linear function. Therefore, the correct match is B.

The equation 3x - 4y = 27 can be rearranged to solve for y in terms of x: y = (3/4)x - (27/4). This equation is of the form y = mx + b, where m = 3/4 and b = -27/4, which indicates a linear function. Therefore, the correct match is C.

The equation 3x² - 2x = 4y is a quadratic equation, which has a variable raised to the second power. Quadratic functions are non-linear, so the correct match is D.

The graph shows a straight line, which indicates a linear function. We can determine the equation of the line by finding the slope and y-intercept. The slope is (5 - (-1))/(2 - (-2)) = 6/4 = 3/2, and the y-intercept is -1. So the equation of the line is y = (3/2)x - 1, which is a linear function. Therefore, the correct match is E.

The table of values does not show a constant rate of change, which indicates a non-linear function. For example, between x = -2 and x = -1, y changes by 5, but between x = 0 and x = 1, y changes by only 1. This suggests that the function is not linear. We can't determine the exact form of the function from the table alone, but we can conclude that it is a non-linear function. Therefore, the correct match is F.

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Answer:

45/100, 0.45, 9/20

Step-by-step explanation:

Answer 1:  A percentage is part of a number is written in hundredths.  Thus, 45% is the same as 45/100

Answer 2:  To convert 45% to a decimal, you can rewrite 45% as 45.0%.  Then, you can move the decimal two places to the left, which is the same as dividing by 100.  You can also do standard division and you'll find the 45 divided by 100 = 0.45

Answer 3:  45/100 is not the simplest form of this fraction.  To simplify, we must find the greatest common factor of the numerator (45) and the denominator (100).  Both of these numbers can be divided evenly by 5 so 5 is the GCF of the two numbers.

Now, we divide both numbers by 5 to simplify the fraction:

45 / 5 = 9 and 100 / 5 = 20.  Thus, 45/100 reduces down to 9/20

Select any of the two answers available on your worksheet (if I'm reading what you wrote correctly, it seems like the two answer choices they provided were 9/20 and 45/100

The area of the square is equal to 64 square units.

In this problem we find the representation of a geometric system formed by a circle and a square and we must compute the area of the latter figure. First, find the side length of the square by Pythagorean theorem:

10² = (2 · x)² + (1.5 · x)²

100 = 4 · x² + 2.25 · x²

100 = 6.25 · x²

x² = 16

x = 4

Second, compute the area of the square:

A = (2 · 4)²

A = 64

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Answer:

Step-by-step explanation:

Center of circle will be at the midpoint of (-2,1) and (4,3):

                     [tex]M=(\frac{x_1+x_2}{2} ,\frac{y_1+y_2}{2} )[/tex]

                           [tex]=(\frac{-2+4}{2} ,\frac{1+3}{2} )[/tex]

                           [tex]=(1,2)[/tex]

Radius will be distance from circle center (1,2) to edge (4,3):

                       [tex]r=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2 }[/tex]

                           [tex]=\sqrt{(4-1)^2+(3-2)^2 }[/tex]

                           [tex]=\sqrt{3^2+1^2 }[/tex]

                           [tex]=\sqrt{10}[/tex]

Equation of circle with center (1,2) and radius [tex]\sqrt{10}[/tex]:

                      [tex](x-h)^2+(y-k)^2=r^2[/tex]        center (h,k)  radius r

                      [tex](x-1)^2+(y-2)^2=\sqrt{10}^2[/tex]

                      [tex](x-1)^2+(y-2)^2=10[/tex]

SOLUTION: [tex](x-1)^2+(y-2)^2=10[/tex]

Yes,I would have made the same decision to purchase disability insurance .

What is disability insurance?

Disability insurance provides financial protection in the event that you become unable to work due to an illness or injury. It can help you pay for living expenses, medical bills, and other costs while you are unable to work.

When deciding whether or not to purchase disability insurance, it's important to consider your financial situation and your ability to manage expenses if you were to become disabled. If you have a good amount of savings or other sources of income, you may be able to manage without disability insurance. However, if you rely heavily on your income to pay for living expenses and do not have significant savings, disability insurance may be a wise investment.

It's also important to consider the cost of disability insurance and whether it fits within your budget. While the additional $22.70 a month may not seem like a lot, it can add up over time. You should evaluate whether the cost of the insurance is worth the potential benefits it provides.

Ultimately, the decision to purchase disability insurance is a personal one that depends on individual circumstances and priorities. While it's impossible to predict the future, having disability insurance can provide peace of mind and financial protection in case the unexpected happens.

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Answer:D=12, R=6

A=πr², C=2πr

A=36π, C=12π

A=113.04 ft²

C=37.68 ft

In the given figure the similarity is angle-angle property.

An angle is the result of the intersection of two lines.

An "angle" is the length of the "opening" between these two beams.

Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.

In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.

The two primary components of an angle are the limbs and the vertex.

The joint vertex is the common terminal of the two beams.

Hence, In the given figure the similarity is angle-angle property.

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To document the total daily amount of liquid medication using the metric system, we need to convert teaspoons to milliliters, which is the standard unit of volume in the metric system.

One teaspoon is equal to approximately 5 milliliters, so three teaspoons would be equal to 15 milliliters.

If the patient is taking 15 milliliters every four hours, then they would be taking the medication 6 times a day (24 hours divided by 4 hours per dose).

Therefore, the total daily amount of the medication would be:

15 milliliters per dose x 6 doses per day = 90 milliliters per day

So, the total daily amount of the liquid medication should be documented as 90 milliliters per day in the patient's medical records using the metric system.

The area of the unshaded region is 3x²-3x-8

What is area of rectangle?

A rectangle is a closed 2-D shape, having 4 sides, 4 corners, and 4 right angles (90°). The opposite sides of a rectangle are equal and parallel.

The area of a rectangle is expressed as;

A = l× w

Area of the unshaded part = Area of the whole rectangle - area of shaded part

Therefore area of the unshaded part =

5x²+4x-8 -(2x²+7x)

= 5x²+4x-8 -2x²-7x

collect like terms

5x²-2x²+4x-7x-8

= 3x²-3x-8

therefore the area of the unshaded part is 3x²-3x-8

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Answer:

y = -7x - 3

Step-by-step explanation:

The slope-intercept form of the equation of a line is:

y = mx + b where m is the slope and b is the y-intercept.

Given the slope m = -7 and the y-intercept b = -3, we can write the equation of the line in slope-intercept form as:

y = -7x - 3

Therefore, the equation of the line with slope -7 and y-intercept -3 in slope-intercept form is y = -7x - 3.

Answer: We can start by using the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.

So, we can set each factor equal to zero and solve for theta:

Factor 1: cot(theta) - 1 = 0

Using the identity cot(theta) = cos(theta) / sin(theta), we can rewrite this as:

cos(theta) / sin(theta) - 1 = 0

cos(theta) - sin(theta) = 0

cos(theta) = sin(theta)

Dividing both sides by cos(theta), we get:

tan(theta) = 1

The solutions to this equation are theta = 45 degrees and theta = 225 degrees (since tangent is positive in the first and third quadrants).

Factor 2: 2sin(theta) + sqrt(3) = 0

Subtracting sqrt(3) from both sides and dividing by 2, we get:

sin(theta) = -sqrt(3)/2

The solutions to this equation are theta = 240 degrees and theta = 300 degrees (since sine is negative in the third and fourth quadrants, and sin(240) = sin(300) = -sqrt(3)/2).

Therefore, the exact solutions in the interval [0 degrees, 360 degrees) are:

theta = 45 degrees, 225 degrees, 240 degrees, 300 degrees.

Step-by-step explanation:

According to the information to graph the inequality we have to show a number line with an open circle at -3 and an arrow pointing to the left, with the shaded region to the left of -3 labeled as x < -3.

To graph the inequality x < -3, we need to draw a number line and mark the point -3 on it. Since the inequality is "less than," we want to shade the region to the left of -3 on the number line.

We start by drawing a horizontal line with a labeled point for -3. Then we draw an arrow to the left of -3, to indicate that all the values to the left of -3 satisfy the inequality.

Here are the steps to graph the inequality on the sketchpad:

The resulting graph should show a number line with an open circle at -3 and an arrow pointing to the left, with the shaded region to the left of -3 labeled as x < -3.

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Answer:

Step-by-step explanation:

A) To find the area covered by two-way radio A, we need to calculate the area of a circle with a radius of 8 miles:

Area = πr² = 3.14 x 8² = 200.96 square miles

Rounding to the nearest whole number, two-way radio A covers 201 square miles.

B) To find the area covered by two-way radio B, we need to convert the radius from kilometers to miles and then calculate the area of a circle with that radius:

Radius in miles = 11.27 / 1.61 = 6.9988 miles (rounded to 4 decimal places)

Area = πr² = 3.14 x (6.9988)² = 153.94 square miles

Rounding to the nearest whole number, two-way radio B covers 154 square miles.

C) Two-way radio A covers a larger area than two-way radio B (201 square miles vs 154 square miles).

D) The scale factor relationship between the radio coverages can be found by dividing the radius of radio A by the radius of radio B:

Scale factor = radius of A / radius of B = 8 miles / (11.27 km / 1.61 km/mile) = 4.97

This means that the coverage of two-way radio A is almost 5 times larger than that of two-way radio B.

a. Sm satisfies all three properties of an equivalence relation, it is indeed an equivalence relation on Real numbers[x].

b. The equivalence class of Sm containing f is the set of all polynomials g that satisfy the condition m | (g - f).

c. The equivalence class of Sm containing the polynomial f(x) = x + 1 is the set {g ∈ Real numbers[x] : g(2) = 3}.

Real numbers are those that can be used to calculate constant values like temperature, time, or distance. They are equivalent to countless decimal increases. They are the sum of all positive and negative integers, fractions, decimals, transcendental numbers, and irrational numbers.

(a) To prove that Sm is an equivalence relation on Real numbers[x], we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.

Reflexivity: For any polynomial f ∈ Real numbers[x], we have f − f = 0, which is clearly a multiple of any polynomial m. Therefore, (f, f) ∈ Sm for all f ∈ Real numbers[x], and Sm is reflexive.

Symmetry: If (f, g) ∈ Sm, then m is a factor of g − f. This means that there exists a polynomial q ∈ Real numbers[x] such that g − f = mq. It follows that f − g = −mq, which is a multiple of m. Therefore, (g, f) ∈ Sm, and Sm is symmetric.

Transitivity: If (f, g) ∈ Sm and (g, h) ∈ Sm, then m is a factor of g − f and m is a factor of h − g. This means that there exist polynomials q1 and q2 ∈ Real numbers[x] such that g − f = mq1 and h − g = mq2. It follows that h − f = (h − g) + (g − f) = mq2 + mq1 = m(q2 + q1). Therefore, m is a factor of h − f, and (f, h) ∈ Sm. Thus, Sm is transitive.

Since Sm satisfies all three properties of an equivalence relation, it is indeed an equivalence relation on Real numbers[x].

(b) The division rule for polynomials states that for any polynomials f and g with g ≠ 0, there exist unique polynomials q and r such that f = qg + r and deg r < deg g. The equivalence class of Sm containing f is the set of all polynomials g that satisfy the condition m | (g - f).

Applying the division rule for polynomials to the polynomial g - f, we can write it as g - f = mq + r, where deg r < deg m. This means that g - r = f + mq, and therefore, any polynomial in the equivalence class of Sm containing f can be written as g = r + f + mq, where deg r < deg m. In other words, every equivalence class of Sm contains a polynomial of the form r + f + mq, where r is a polynomial of degree less than deg m.

(c) Let m(x) = x - 2. Then for any polynomial f(x) ∈ Real numbers[x], we have (f, g) ∈ Sm if and only if g(x) - f(x) is a multiple of x - 2. This means that g(2) - f(2) = 0, or in other words, f(2) = g(2). Therefore, the equivalence class of Sm containing the polynomial f(x) = x + 1 is the set {g ∈ Real numbers[x] : g(2) = 3}.

Justification: Let g(x) = (x - 2) + 3. Then g(2) = 3, and g(x) - f(x) = (x - 2) + 3 - (x + 1) = x - 2, which is a multiple of x - 2. Therefore, (f, g) ∈ Sm, and the set {g ∈ Real numbers[x] : g(2) = 3} is an equivalence class of Sm.

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Accοrding the given questiοn the x- and y- intercepts, are as fοllοws,

x-intercept =(28, 0), y-intercept = (0, -4)

An intercept is the pοint οf intersectiοn οf the line with the cοοrdinate axes. If the line intersects with the X axis, then the y cοοrdinate οf the pοint οf intersectiοn is zerο and the intercept is called the x intercept.

Tο find the x-intercept, we need tο sοlve fοr x when y = 0:

0 = lοg2(x + 2) - 5

5 = lοg2(x + 2)

[tex]2^5 = x + 2[/tex]

x = 30 - 2 = 28

Therefοre, the x-intercept is (28, 0).

Tο find the y-intercept, we need tο evaluate the functiοn when x = 0:

f(0) = lοg2(0 + 2) - 5

f(0) = lοg2(2) - 5

f(0) = 1 - 5

f(0) = -4

Therefοre, the y-intercept is (0, -4)

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[tex] \:\:\:\:\:\:\star\longrightarrow \sf \underline{y = 5\:x^4\:tanx}\\[/tex]

Differentiating with respect to x-

[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{d}{dx}\:y = \dfrac{d}{dx}\: 5\:x^4\:tanx\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}\:= 5\:\dfrac{d}{dx}\: \:x^4\:tanx\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{d}{dx}\:y =5\:\bigg[tanx\times \dfrac{d}{dx}\: \:x^4\:+x^4\times \dfrac{d}{dx} tanx\bigg]\\[/tex]

[tex]\pink{\star}\:\boxed{\sf\sf\dfrac{d}{dx}\bigg[f(x)\:g(x)\bigg] = f(x) \dfrac{d}{dx} g(x) + g(x) \dfrac{d}{dx} f(x)}\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}= 5\:\bigg[tanx\times 4\:x^3+ x^4 \times sec^2x \bigg]\\[/tex]

[tex] \qquad\pink{\star}\:\:\:\:\boxed{\sf \dfrac{d}{dx}x^n=nx^{n-1}}\\[/tex]

[tex] \qquad\pink{\star}\:\:\:\:\boxed{\sf \dfrac{d}{dx}tanx=sec^2x}\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf\dfrac{dy}{dx}= 5\:x^3\:\bigg[4tanx+ x sec^2x \bigg]\\[/tex]

[tex] \:\:\:\:\:\:\longrightarrow \sf\underline{\dfrac{dy}{dx}= \boxed{\sf5\:x^3\:\bigg[4tanx+ x sec^2x \bigg]}}\\[/tex]

The median amount of time Julie's friends spend reading each week is 3 1/2 hours, which is the average of the two middle values when the given values are arranged in order.

To find the median value, we first need to put the given values in order from smallest to largest. The given values are: 1/2, 1/2, 2 1/2, 4 1/2, 7, and 12. When we arrange them in order, we get:

1/2, 1/2, 2 1/2, 4 1/2, 7, 12

The median is the middle value of the ordered list. If there is an odd number of values, the median is simply the middle value. However, in this case, we have an even number of values. Therefore, we take the average of the two middle values to find the median. The two middle values are 2 1/2 and 4 1/2. So, the median is:

(2 1/2 + 4 1/2)/2 = 3 1/2

This means that half of Julie's friends read for more than 3 1/2 hours per week and half of them read for less than 3 1/2 hours per week.

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Answer:

Use in² for area; use in³ for volume.

Explanation:

When you calculate the area of a rectangle, you multiply length times width. Both units are in inches so:

inches × inches = (inches)² or in²

When you calculate the volume of a box (rectangular prism), you multiply length by width by height. All three units are in inches so:

inches × inches × inches = (inches)³ or in³

The probability that more than 2 students have blue eyes is 0.149, the expected number of students out of the 4 with blue eyes is 0.4, the variance is 0.36, and the standard deviation is 0.6.

a) P(X = 3) = (4 choose 3) * [tex](0.10)^3 * (0.90)^1[/tex] = 0.0036

So the probability that exactly 3 of them have blue eyes is 0.0036.

b) P(X = 2) = (4 choose 2) * [tex](0.10)^2 * (0.90)^2[/tex] = 0.0386

So the probability that exactly 2 of them have blue eyes is 0.0386.

c) P(at least 1) = 1 - P(none) = 1 - [tex](0.90)^4[/tex] = 0.3439

So the probability that at least 1 of them has blue eyes is 0.344.

d) P(X > 2) = P(X = 3) + P(X = 4) = (4 choose 3) * [tex](0.10)^3 * (0.90)^1[/tex] + (4 choose 4) * [tex](0.10)^4 * (0.90)^0[/tex] = 0.0001

So the probability that more than 2 have blue eyes is 0.0001.

e) E(X) = n * p = 4 * 0.10 = 0.4

So the expected number of students out of the 4 with blue eyes is 0.4, not 0.291 as provided.

f)Var(X) = n * p * (1 - p) = 4 * 0.10 * 0.90 = 0.36

So the variance is 0.36.

g) SD(X) = sqrt(Var(X)) = sqrt(0.36) = 0.6

Standard deviation is 0.6.

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Answer:

Step-by-step explanation:

I'm a little confused where the angle is, because I don't understand English units of measurement well, but the answers must be correct!

i use formula  [tex]sin^{2} \alpha + cos^{2} \alpha =1[/tex]

Answer: Let's break down the budget into different categories:

Furniture: 15% of the budget

15% of Rs. 53000 = Rs. 7950

So you can spend Rs. 7950 on furniture.

Cosmetics: 25% of the budget

25% of Rs. 53000 = Rs. 13250

So your cosmetics bill from Makeup4uonline is Rs. 13250.

Appliances: 15% of the budget at 20% discount

15% of Rs. 53000 = Rs. 7950

The appliances are at a 20% discount, so you only need to pay 80% of the price:

80% of Rs. 7950 = Rs. 6360

So you can spend Rs. 6360 on appliances.

Electric works: 5% of the total budget which is 12% of wall décor and lights

12% of the budget = 12/100 x Rs. 53000 = Rs. 6360

5% of the budget = 5/100 x Rs. 53000 = Rs. 2650

So you can spend Rs. 6360 on wall décor and lights, and Rs. 2650 on electric works.

Leftover amount for COVID-19 safety measures

The total amount spent so far is:

Rs. 7950 + Rs. 13250 + Rs. 6360 + Rs. 2650 = Rs. 30110

So the leftover amount for COVID-19 safety measures is:

Rs. 53000 - Rs. 30110 = Rs. 22890

Rate of leftover amount to total budget

The rate of leftover amount to total budget is:

(Rs. 22890 / Rs. 53000) x 100% = 43.2%

So 43.2% of the total budget is leftover for COVID-19 safety measures.

Step-by-step explanation:

the equation [tex]r = 3 - 3cos(\theta)[/tex] is symmetric with respect to the vertical line  but not with respect to the origin or the pole .

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.  

The equation r = 3 - 3cos(θ) represents a polar curve, where r is the distance from the origin to a point on the curve, and θ is the angle between the positive x-axis and the line connecting the origin to the point.

1. Symmetry with respect to the origin: A curve is symmetric with respect to the origin if replacing (r, θ) with (-r, θ + π) gives the same curve. Substituting -r for r and θ + π for θ in the equation, we get:

[tex]-r = 3 - 3cos(\theta + \pi)[/tex]

[tex]-r = -3 - 3cos(\theta)[/tex]

[tex]r = 3 + 3cos(\theta)[/tex]

This is not the same as the original equation, so the curve is not symmetric with respect to the origin.

2.Symmetry with respect to the pole (x-axis): A curve is symmetric with respect to the pole if replacing (r, θ) with (r, -θ) gives the same curve. Substituting -θ for θ in the equation, we get:

[tex]r = 3 - 3cos(-\theta)[/tex]

[tex]r = 3 + 3cos(\theta)[/tex]

This is not the same as the original equation, so the curve is not symmetric with respect to the pole.

3.Symmetry with respect to the vertical line (y-axis): A curve is symmetric with respect to the vertical line if replacing (r, θ) with (-r, π - θ) gives the same curve. Substituting -r for r and π - θ for θ in the equation, we get:

[tex]-r = 3 - 3cos(π - \theta)[/tex]

[tex]-r = -3 + 3cos(\theta)[/tex]

[tex]r = 3 - 3cos(\theta)[/tex]

This is the same as the original equation, so the curve is symmetric with respect to the vertical line (y-axis).

Therefore, the equation  [tex]r = 3 - 3cos(\theta)[/tex] is symmetric with respect to the vertical line (y-axis) but not with respect to the origin or the pole (x-axis).

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The graph of r = 3 - 3cos(theta) is symmetric about the x-axis but not about the origin or about the y-axis.

Symmetry is described in geometry as a balanced and proportionate likeness seen in two sides of an object. It signifies that one side is a mirror image of the other. The line of symmetry is the imaginary line or axis along which you can fold a figure to obtain the symmetrical halves.

The symmetry equation is r = 3 -3 cosФ depicts a cardioid-shaped polar graph. We can use the following tests to determine symmetry:

1) We can test for symmetry with regard to the origin by seeing if the equation remains the same after substituting r with -r and Ф with Ф + pi.

r = 3 - 3cosФ - r = 3 - 3cos(Ф + pi)

3 - 3(-cos(Ф)) = r

-r = 3 + 3cos(Ф)

Because this is not equal to the original equation, the graph is not symmetric about the origin.

2) Symmetry with respect to the x-axis: We may test for symmetry with respect to the x-axis by seeing if the equation stays unchanged when theta is replaced with -theta.

r = 3 - 3cos(Ф)

r = 3 - 3cos(-Ф)

r = 3 - 3cos(Ф)

This is the same as the original equation, hence the graph is symmetric about the x-axis.

3) Symmetry with respect to the y-axis: We may test for symmetry with respect to the y-axis by seeing if the equation stays unchanged when theta is replaced with pi - theta.

r = 3 - 3cos(Ф)

r = 3 - 3cos(pi - Ф)

r = 3 + 3cos(Ф)

Because this is not equal to the original equation, the graph is not symmetric about the y-axis.

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Answer:

X=35

Step-by-step explanation:

if the translated part is the full question; the answer is attached ⬆️.

Hope it helps