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Give the degree of the polynomial.
w³x²-14x^9v²w² +5−2v
0
8.9
X

Angle B is 50 degrees in measurement.

One method to find the measure of angle b is to use the properties of angles in a triangle. We know that the sum of the angles in a triangle is 180 degrees. We also know that angles a and c have measures of 50 degrees and 80 degrees respectively. Therefore, we can find the measure of angle b by subtracting the sum of angles a and c from 180 degrees:

angle b = 180 degrees - angle a - angle c

angle b = 180 degrees - 50 degrees - 80 degrees

angle b = 50 degrees

Therefore, angle b has a measure of 50 degrees. Another method to find the measure of angle b is to use trigonometry, such as the sine or cosine rule, depending on the given information.

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What is one method to find the measure of angle B?

A. use the Pythagorean theorem to find BC, then solve the equation tan(B)=8/BC

B. because of the 30-60-90 triangle theorem, you know the measure of angle B is 60

C. solve the equation cos(B)=8/89(square rooted)

The expression 4x^2 + 14x + 10 can be completely factored into 2(2x + 5)(x + 1).

We can begin by factoring out the biggest common factor of the three components, which is 2, in order to factor 4x2 + 14x + 10 completely:

4x^2 + 14x + 10 = 2(2x^2 + 7x + 5)

The quadratic expression 2x2 + 7x + 5 must now be factored. Finding two binomials whose product is equal to 2x2 + 7x + 5 will help us achieve this.

Choose two integers that sum up to 7 and multiply by 2*5 to factor the expression. These are the digits 2 and 5. Hence, we can write:

2x^2 + 7x + 5 = 2x^2 + 2x + 5x + 5

= 2x(x + 1) + 5(x + 1)

= (2x + 5)(x + 1)

Adding this to our initial expression yields the following:

4x^2 + 14x + 10 = 2(2x + 5)

(x + 1)

As a result, the phrase 2(2x + 5)(x + 1) may completely factor the expression 4x2 + 14x + 10.

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The relationship between 30 hours and 15 hours is that the number of hours spent using electronic devices is twice the number of hours of time spent playing sports, i.e. 30 hours = 2 x 15 hours.

There are different ways to approach this question, but one possible relationship between 30 hours and 15 hours to complete the statement "the number of hours students spent using electronic devices is times the number of hours spent playing sports" is:

If a student spends 30 hours using electronic devices and 15 hours playing sports, then the number of hours spent using electronic devices is twice the number of hours spent playing sports.

We can express this relationship using variables as follows:

Let E be the number of hours spent using electronic devices, and let S be the number of hours spent playing sports. Then, we can write:

E = 2S

If we substitute 30 for E and 15 for S in this equation, we get

30 = 2(15)

This equation is true, which means that the relationship holds for these particular values of E and S.

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The range of values of x that satisfy sec(x) = -√(2) and π/2 is:

x = 3π/4, 5π/4, 0.453, 5.829.

The range of values of x is calculated as follows;

Since sec(x) = 1/cos(x), we can use the identity cos^2(x) + sin^2(x) = 1 to solve for cos(x).

First, we consider the case where sec(x) = -√(2).

We know that sec(x) = 1/cos(x), so we can write:

1/cos(x) = -√(2)

Multiplying both sides by cos(x) gives:

1 = -√(2)cos(x)

Dividing both sides by -√(2) gives:

-1/√(2) = cos(x)

So, x is an angle whose cosine is -1/√(2). This occurs in the second quadrant, where cosine is negative. We can find the reference angle for this value of cosine by taking the arccosine of its absolute value:

arccos(|-1/√(2)|) = π/4

Therefore, x is either:

x = π - π/4 = 3π/4

or

x = π + π/4 = 5π/4

Next, we consider the case where sec(x) = π/2. We know that sec(x) = 1/cos(x), so we can write:

1/cos(x) = π/2

Multiplying both sides by cos(x) gives:

1 = π/2 cos(x)

Dividing both sides by π/2 gives:

2/π = cos(x)

So, x is an angle whose cosine is 2/π. This occurs in the first quadrant, where cosine is positive. We can find the reference angle for this value of cosine by taking the arccosine:

arccos(2/π)

Using a calculator, we find that:

arccos(2/π) ≈ 0.453

Therefore, x is either:

x = 0.453

or

x = 2π - 0.453 ≈ 5.829

So the range of values of x that satisfy sec(x) = -√(2) and π/2 is:

x = 3π/4, 5π/4, 0.453, 5.829

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The complete question is below:

If sec(x) = -√(2) and π/2, find the range of values of x

The probability that between 44% and 54% of the sampled adults work during summer vacation is approximately 1.

The most important continuous probability distribution in probability theory and statistics is the normal distribution, often known as the gaussian distribution. It is also known as a bell curve sometimes. In every physical field and in economics, the normal distribution accurately or nearly represents a huge number of random variables. Moreover, it may be used to approximate different probability distributions, supporting the employment of the name "normal" as in reference to the most common distribution.

Given that, adults that work during summer vacation is:

p=0.49

The mean is:

μ = np = (300)(0.49) = 147

The standard deviation is given by:

σ = sqrt(npq) = sqrt((300)(0.49)(0.51)) ≈ 8.24.

Now, the probability that between 44% and 54%:

z1 = (0.44 - 0.49) / 0.00824 ≈ -6.07

z2 = (0.54 - 0.49) / 0.00824 ≈ 6.07

The area under this curve using the z-table is 1.

Thus, probability that between 44% and 54% of the sampled adults work during summer vacation is approximately 1.

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The coordinates of B of the diameter of the circle is: (5, -1).

Since AB is a diameter of the circle, its midpoint will be the center of the circle, which is given to be (1, 1). Therefore, we can find the coordinates of point B by using the midpoint formula.

Midpoint formula:

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2).

Let A be (-3, 3), which is one endpoint of the diameter AB. Let B be the other endpoint of the diameter AB.

Since the midpoint of AB is (1, 1), we have:

((x-coordinate of A + x-coordinate of B)/2, (y-coordinate of A + y-coordinate of B)/2) = (1, 1)

Substituting the coordinates of point A, we get:

((-3 + x-coordinate of B)/2, (3 + y-coordinate of B)/2) = (1, 1)

Multiplying both sides of each equation by 2, we get:

(-3 + x-coordinate of B, 3 + y-coordinate of B) = (2, 2)

Adding 3 to both sides of the first equation and subtracting 3 from both sides of the second equation, we get:

(x-coordinate of B, y-coordinate of B) = (2+3, 2-3) = (5, -1)

Therefore, the coordinates of point B are (5, -1).

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Answer:

Y=3

Step-by-step explanation:

Put in x=-1

y-3 = 2(-1+1)

y-3 = 2(0)

y-3=0

add 3 more to both sides

y-3+3 = 0+3

y =3

(Don't forget Brainliyest)

Answer:

[tex] \sf \: y = 3[/tex]

Step-by-step explanation:

Now we have to,

→ Find the required value of y.

We have to use,

→ x = -1

The equation is,

→ y - 3 = 2(x + 1)

Then the value of y will be,

→ y - 3 = 2(x + 1)

→ y = 2(x + 1) + 3

→ y = 2((-1) + 1) + 3

→ y = 2(0) + 3

→ y = 0 + 3

→ [ y = 3 ]

Hence, the value of y is 3.

The length of the rectangle is 15 dm and the width is 13 dm.

Let's assume that the length of the rectangle is "L" and the width is "W".

From the problem statement, we have two pieces of information:

The area of the rectangle is 195 dm²:

Area = Length x Width

195 dm² = L x W

The width is two less than the length:

W = L - 2

Now, we can substitute the second equation into the first equation to eliminate W and get an equation with only one variable:

195 dm² = L x (L - 2)

Simplifying the equation:

195 dm² = L² - 2L

L² - 2L - 195 dm² = 0

To solve for L, we can use the quadratic formula:

L = (-b ± √(b² - 4ac)) / 2a

Where a = 1, b = -2, and c = -195.

L = (2 ± √(2² + 4 x 1 x 195)) / 2 x 1

L = (2 ± √4 + 780) / 2

L = (2 ± √784) / 2

L = (2 ± 28) / 2

L = 15 or L = -13

Since the length can't be negative, the length of the rectangle is L = 15 dm.

Now we can use the equation W = L - 2 to find the width:

W = 15 dm - 2 dm

W = 13 dm

Therefore, the length of the rectangle is 15 dm and the width is 13 dm.

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Therefore, the probability that st > 0 is 9/24 or 3/8, which is approximately 0.375 or 37.5%.

Probability is a measure of how likely an event is to occur. It is a number between 0 and 1, with 0 suggesting that an occurrence is impossible and 1 indicating that an event is unavoidable. A given event's probability is computed by dividing the number of positive outcomes by the total number of potential possibilities.

Here,

To find the probability that st > 0, we need to consider all possible pairs of values (s, t) such that their product is positive.

We can start by considering the possible pairs of values for s and t separately.

For s, there are three possible values that are negative: -4, -3, and -1. There are also three possible values that are positive or zero: 0, 2, and 8.

For t, there are two possible values that are negative: -7 and 1. There are also two possible values that are positive: 4 and 6.

We can now list all possible pairs of values (s, t) and determine whether their product is positive:

(-4, -7): Negative

(-4, 1): Negative

(-4, 4): Negative

(-4, 6): Negative

(-3, -7): Positive

(-3, 1): Negative

(-3, 4): Negative

(-3, 6): Negative

(-1, -7): Positive

(-1, 1): Negative

(-1, 4): Negative

(-1, 6): Negative

(0, -7): Negative

(0, 1): Zero

(0, 4): Zero

(0, 6): Zero

(2, -7): Negative

(2, 1): Positive

(2, 4): Positive

(2, 6): Positive

(8, -7): Negative

(8, 1): Positive

(8, 4): Positive

(8, 6): Positive

Out of the 24 possible pairs, there are 9 pairs whose product is positive.

P=9/24

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Answer:

Below

Step-by-step explanation:

See image below

The value of the 3rd term in the expansion of (x+6) ^6 will be as follows:

540x^4

What is expansion?

Expanding brackets, also known as multiplying out, seeks to eliminate the set of brackets by multiplying each phrase inside a bracket by the term on the outside and subsequently accumulating similar phrases. When solving equations, extending brackets, which is the opposite of factorization, is frequently an essential step.

Here in the question,

We have,

(x+6) ^6

Expanding it we get:

= x^6 + 36x^5 + 540x^4 +4320x³ + 19440x² + 46656x + 46656

So, the 3rd term of the expansion is 540x^4.

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The value of the 3rd term in the expansion of [tex](x+6) ^6[/tex] will be as follows: [tex]540x^4[/tex]. The correct answer is option (c).  [tex](6\ \ 4)36x^4[/tex]

By multiplying each phrase inside a bracket by the word on the outside and then accumulating similar phrases, expanding brackets, also known as multiplying out, aims to eliminate the set of brackets. Extending brackets, which is the opposite of factorization, is frequently a crucial stage in the solution of equations.

An affine transformation termed expansion, in which the scale is expanded, is also referred to as an enlargement or dilation. It is also sometimes referred to as an enlargement and is the polar opposite of a geometric constriction.

Here in the question,

We have,

[tex](x+6) ^6[/tex]

Expanding it we get:

[tex]= x^6 + 36x^5 + 540x^4 +4320x^3 + 19440x^2 + 46656x + 46656[/tex]

So, the 3rd term of the expansion is [tex]540x^4[/tex].

The correct answer is option (c). [tex](6\ \ 4)36x^4[/tex]

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The ramp will rise 18 inches from the ground at its highest end.

To determine the height of the ramp at its highest end, we can use trigonometry and the given angle of elevation.

In a right triangle formed by the ramp, the ground, and the height of the ramp, the angle of elevation (30 degrees) is the angle between the ground and the hypotenuse (the ramp itself). The height of the ramp is the opposite side, and the length of the ramp is the hypotenuse.

Using the trigonometric function sine (sin), we can set up the equation:

sin(30 degrees) = opposite/hypotenuse

sin(30 degrees) = height/36 inches

Since the sine of 30 degrees is 0.5:

0.5 = height/36 inches

To solve for the height, we can multiply both sides of the equation by 36:

0.5 x 36 inches = height

18 inches = height

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The answer is (b) 24 students thought they watched too much television.

The solution of the given question are as following :-

The given graph represents the responses of 120 students who were surveyed about their television viewing habits. The students were asked whether they spent too little, about the right amount, or too much time watching television, or whether they didn't know.

Out of the total 120 students surveyed, 30% thought that they spent too little time watching television, while only 5% felt that they spent about the right amount of time. A further 20% felt that they spent too much time watching television, while the remaining 45% didn't know.

To answer the question of how many students thought they watched too much television, we need to focus on the 20% who said that they spent too much time watching TV. This percentage can be converted to a whole number by multiplying it with the total number of students surveyed, which is 120.

20/100 x 120 = 24

Therefore, 24 students out of 120 thought that they watched too much television.

The survey results indicate that a significant proportion of students, 50% (30% who thought they watched too little and 20% who thought they watched too much), felt that they were not watching the right amount of television. This suggests that there may be a need for students to be more mindful of their television viewing habits and make adjustments accordingly.

It's also worth noting that nearly half of the surveyed students, 45%, were unsure about how much television they watched. This could be because they don't pay attention to the amount of time they spend watching TV or because they have a hard time evaluating whether their television viewing habits are appropriate.

Overall, the survey results highlight the importance of being mindful of how much time we spend watching television and making sure that we are not spending too much time on it. It's also essential to evaluate whether our television viewing habits align with our personal preferences and priorities.

The calculation part is as follows :-

Out of 120 students:

30% thought they watched too little television, which is 30/100 x 120 = 36 students.

5% thought they watched about the right amount of television, which is 5/100 x 120 = 6 students.

20% thought they watched too much television, which is 20/100 x 120 = 24 students.

45% didn't know how much television they watched, which is 45/100 x 120 = 54 students.

Therefore, the answer is (b) 24 students thought they watched too much television.

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The speed of the current is approximately 39.05 miles per hour.

Distance equals rate times time in the equation for distance, rate, and time. This equation shows how far an item moves over a specific amount of time at a specific pace by relating the three variables in a linear equation. Any of the three variables can be solved for by rearranging the formula.

Let us suppose the speed of current = c.

Then the formula for distance is given as:

distance = rate x time

For upstream:

distance = 48 miles

rate = 20 - c miles per hour

time = distance / rate

= 48 / (20 - c) hours

For downstream:

distance = 48 miles

rate = 20 + c

time = distance / rate

= 48 / (20 + c) hours

The total time for the trip is 5 hours thus,

48 / (20 - c) + 48 / (20 + c) = 5

Taking the LCM:

48(20 + c) + 48(20 - c) = 5(20 - c)(20 + c)

1920 = 400 - c²

c² = 1520

c ≈ 39.05 miles per hour

Hence, the speed of the current is approximately 39.05 miles per hour.

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The probability οf getting 6 and 2 = [tex]\frac{1}{18}[/tex]

There are 216 ways to roll 3 dice.

Prοbability is a way of calculating how likely something is to happen. It is difficult to prοvide a complete prediction for many events. Using it, we can οnly forecast the probability, or likelihood, of an event occurring. The prοbability might be between 0 and 1, where 0 denotes an impοsibility and 1 denotes a certainty.

Here  that dice has been rοlled twice. So, there could be 1 number on the top side each time.

I also assume that the numbers may οr may not appear in given sequence.

Prοbability of getting 6 or 2 on first time = [tex]\frac{2}{6}[/tex]

Now since we have gοt either of six or two we need the other one now.

Examples are better for explaining. Sο, consider that two appeared on top now we need 6.

Prοbability of getting 6 = [tex]\frac{1}{6}[/tex]

Then prοbability of rοlling 6 and 2 = [tex]\frac{2}{6}\times\frac{1}{6}[/tex] = [tex]\frac{1}{18}[/tex]

When a dice is rolled, there are six possible outcomes.

So, the tοtal number of outcomes when three dice are rolled is

=> [tex]6\times6\times6=6^3=216.[/tex]

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Answer:

£155.55

Step-by-step explanation:

To determine the number of British pounds Brian will receive for R2100, begin by calculating the total commission he must pay an agent by multiplying the amount being exchanged (R2100) by the commission rate of 1.5%:

⇒ Commission = R2100 × 0.015 = R31.50

Subtract the commission from the total amount being exchanged to get the net amount:

⇒ Net amount = R2100 - R31.50 = R2068.50

Given R1 is worth 0.075199 British pounds, convert the net amount from South African rand to British pounds by multiplying by the exchange rate:

⇒ British pounds = 2068.50 × 0.075199

⇒ British pounds = £155.55 (2 d.p.)

Therefore, Brian will receive £155.55 for R2100 after paying a commission of R31.50.

Brian will get 155.5501 pounds for 2100 South African rand after paying an agent commission of 1.5%.

To calculate how many pounds Brian will get for 2100 South African rand, we can use the following steps:

Calculate the total amount of pounds that Brian would receive if there were no commission.

To do this, we can multiply the amount of South African rand (2100) by the exchange rate (0.075199):

2100 × 0.075199 = 157.9189 pounds

So without commission, Brian would receive 157.9189 pounds.

Calculate the commission that the agent will charge.

The commission is 1.5% of the total amount, so we can calculate it as:

0.015 × 157.9189 = 2.3688 pounds

So the agent will charge Brian a commission of 2.3688 pounds.

Calculate the final amount of pounds that Brian will receive.

To calculate the final amount of pounds, we can subtract the commission from the total amount of pounds:

157.9189 - 2.3688 = 155.5501 pounds

So Brian will receive 155.5501 pounds after paying the agent commission.

Therefore, Brian will get 155.5501 pounds for 2100 South African rand after paying an agent commission of 1.5%.

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Given function, f(x) = x^6 - 12x^5 + 45x^4 - 405x^2 + 972x - 729; 3 (mult. 5).

Zeroes of f(x) are the values of x for which f(x) = 0. So, f(x) is factorable if and only if we can find zeroes of f(x).

Let's solve f(x) = 0 using x = 3 as the initial guess. Then, f(3) = 3^6 - 12(3^5) + 45(3^4) - 405(3^2) + 972(3) - 729 = 0. So, x = 3 is a zero of f(x) of the given multiplicity, which is 5.

Since x = 3 is a zero of f(x) of multiplicity 5, we can represent f(x) as follows:

$$f(x) = (x-3)^5 p(x)$$

where p(x) = EXPRESSF[X] and EXPRESSF[X] is a polynomial expression in x.

Now, we have to find the polynomial expression p(x) so that we can express f(x) as a product of linear factors.

The best way to find p(x) is by polynomial division:

$$\begin{array}{r|rrrrrr} &x^5&-5x^4&30x^3&-90x^2&180x&-243\\hline x-3&x^6&-12x^5&45x^4&-405x^2&972x&-729\\hline &x^6&-3x^5&+18x^4&-45x^3&135x^2&-243x\ & & & &360x^3&-1080x^2&648x\ & & & &360x^3&-1080x^2&648x\ & & & & &1260x^2&-891x\ & & & & &1260x^2&-3780x\ & & & & & &2889x\\end{array}$$

So, p(x) = x^5 - 3x^4 + 18x^3 - 45x^2 + 135x - 243.

Therefore, we can express f(x) as a product of linear factors as follows:

$$\begin{aligned}f(x) &= (x-3)^5 p(x)\ &= (x-3)^5 (x^5 - 3x^4 + 18x^3 - 45x^2 + 135x - 243)\ &= (x-3)^5 (x-3) (x^4 + 2x^3 + 12x^2 + 36x + 81)\ &= (x-3)^6 (x^4 + 2x^3 + 12x^2 + 36x + 81)\ \end{aligned}$$

Therefore, f(x) is a product of linear factors.

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The values of x and y for the similar triangles are 8m and 9m respectively.

We have the triangles to be similar, this implies that the length EF of the smaller triangle is similar to the length BC of the larger triangle

and the length DF of the smaller triangle is similar to the length AC of the larger triangle

so;

8m/16m = 4m/x

x = (16m × 4m)/8m {cross multiplication}

x = 2 × 4m

x = 8m

y/18m = 8m/16m

y = (18m × 8m)/16m {cross multiplication}

y = 18m/2

y = 9m

Therefore, the values of x and y for the similar triangles are 8m and 9m respectively.

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Answer:

4

Step-by-step explanation:

V = πr²h

h = V/πr² = (314) / π(5)² ≈ 4

We would need 4 pattern block triangles to create 2 trapezoids.

A trapezoid is a four-sided polygon (or quadrilateral) with at least one pair of parallel sides. The parallel sides of a trapezoid are called the bases, and the non-parallel sides are called the legs. The height (or altitude) of a trapezoid is the perpendicular distance between the two bases

To answer this question, we need to know the number of triangles that make up one trapezoid using pattern blocks.

One trapezoid made up of pattern blocks would have the following shapes:

2 trapezoids

2 triangles

1 parallelogram

So, one trapezoid would require 2 triangles.

If we want to create 2 trapezoids, we would need:

2 trapezoids x 2 triangles per trapezoid = 4 triangles

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The additional information that would be needed to prove that △WYZ and △XZY are congruent is (B) WY ≅ ZX.

Triangle congruence: If all three corresponding sides and all three corresponding angles are equal in size, two triangles are said to be congruent.

Slide, twist, flip, and turn these triangles to create an identical appearance.

According to the ASA congruence rule, two triangles are congruent when their two included sides and two included angles are equivalent to each other.

So, we know that:

In △WYZ and △XZY:

ZY = ZY (Common)

∠Z = ∠Y (90°)

Then, additional information could be:

WY ≅ ZX

Therefore, the additional information that would be needed to prove that △WYZ and △XZY are congruent is (B) WY ≅ ZX.

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The statement illustrates the transitive property of congruence, which is a fundamental concept in geometry.

This property states that if two geometric figures are congruent to a third figure, then they are congruent to each other.

In the given statement, ΔABC is congruent to ΔDEF, and ΔDEF is congruent to ΔXYZ. By the transitive property, we can conclude that ΔABC is also congruent to ΔXYZ.

This property is important because it allows us to establish relationships between geometric figures based on their congruence. It is used in many geometric proofs and applications, such as proving theorems, solving problems involving similar triangles, and determining the congruence of geometric shapes.

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Hence time can be 0 to 11 sec when the ball is on the ground.It doesn't go that high.The max height is 123.75 meters, at t = 5.5 seconds.

Any object launched into space with only gravity acting on it is referred to as a projectile. Gravity is the main force affecting a projectile. This doesn't imply that other forces don't affect it; it merely means that their impact is far smaller than that of gravity. A projectile's trajectory is its route after being fired. A projectile is something that is launched or batted, as a baseball.

A golf ball is hit into the air the path of the ball can be described by the equation h=55t-5[tex]t^2[/tex] where h is the height of the ball in meters and t is time in seconds,

When ball is on the ground h=0,

[tex]55t-5t^2=0\\\\t(55-5t)=0[/tex]

t=0 and t=11

Hence time can be 0 to 11 sec when the ball is on the ground.

[tex]55t-5t^2=160\\11t-t^2-32=0\\t^2-11t+32=0\\t=\frac{(11 \ +-\sqrt{11^2-4*1*32}}{2}\\\\t=\frac{11+-\sqrt{7i^2}}{2}\\\\t=\frac{11-+7i}{2}[/tex]

It doesn't go that high.

The max height is 123.75 meters, at t = 5.5 seconds.

The complete question is-

A golf ball is hit in the air. the path of the golf ball can be described by the equation h = 55t - 5t2, where h is the height of the ball in meters and t, is the time after how many seconds will the ball be in 160 high

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The measures of angles 1 and 2 are m∠1 = 50° m∠2 = 130 °

Given that the chord intercepted arc RQ = 53° and the chord intercepted arc ST = 47°, we must determine the angle 1 and angle 2 measurements.

The measure of the angle formed by two chords that intersect within the circle is equal to half the sum of the chord's intercepted arcs, as determined by the geometric property.

Measurement of angle 1 = (53° + 47°)/2 = 100°/2

m∠1 = 50°

m∠2 = 180°

m∠2 = 180° - 50°

m∠2 = 130°

Consequently, m1 = 50° and m2 = 130°.

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Complete Question:

What are the measures of angles 1 and 2? m∠1 = ° m∠2 = °

Answer:

484π or 1520.530844

Step-by-step explanation:

4xπx11²

Answer:

-3x + 3z = 0

Step-by-step explanation:

To find an equation of the plane that contains the curve with the vector equation R(t) = (t, t^3, t), we can use the fact that a plane can be defined by a point and a normal vector to the plane. We can choose any point on the curve as a point on the plane, say (0, 0, 0), and find a normal vector to the plane by taking the cross product of the tangent vectors to the curve at two different points.

To find the tangent vector to the curve at a point (t, t^3, t), we can take the derivative of the vector equation with respect to t:

R'(t) = (1, 3t^2, 1)

So, the tangent vector to the curve at (t, t^3, t) is (1, 3t^2, 1).

Now, we can find the normal vector to the plane by taking the cross product of the tangent vectors at two different points on the curve. Let's choose the points (0, 0, 0) and (1, 1, 1) on the curve:

Tangent vector at (0, 0, 0): R'(0) = (1, 0, 1)

Tangent vector at (1, 1, 1): R'(1) = (1, 3, 1)

The normal vector to the plane is the cross product of these two tangent vectors:

N = R'(0) x R'(1) = (-3, 0, 3)

Now, we can use the point-normal form of the equation of a plane to find the equation of the plane that contains the curve:

N · (r - P) = 0, where N is the normal vector to the plane, P is a point on the plane, and r is a point on the plane.

Substituting in the values we have, we get:

(-3, 0, 3) · (r - (0, 0, 0)) = 0

Simplifying this equation gives us:

-3x + 3z = 0

Therefore, the equation of the plane that contains the curve with the vector equation R(t) = (t, t^3, t) is -3x + 3z = 0.

To solve for x and y, we can use either substitution or elimination method. Let's use the elimination method by multiplying the second equation by 8 and subtracting it from the first equation.

1. Let x be the price of one senior citizen ticket and y be the price of one student ticket. We can set up two equations based on the information given:

4x + 5y = 102

7x + 5y = 126

We can solve for x and y using elimination or substitution. Here is one way to do it using elimination:

Multiply the first equation by -1 to get:

-4x - 5y = -102

Add this equation to the second equation to eliminate y:

3x = 24

Solve for x:

x = 8

Substitute x = 8 into one of the original equations to solve for y:

4(8) + 5y = 102

32 + 5y = 102

5y = 70

y = 14

Therefore, the price of one senior citizen ticket is S8 and the price of one student ticket is S14.

2. Let x be the speed of the plane in still air and y be the speed of the wind. We can set up two equations based on the information given:

x + y = 183

x - y = 141

We can solve for x and y using elimination or substitution. Here is one way to do it using addition:

Add the two equations to eliminate y:

2x = 324

Solve for x:

x = 162

Substitute x = 162 into one of the original equations to solve for y:

162 + y = 183

y = 21

Therefore, the speed of the plane in still air is 162 km/h and the speed of the wind is 21 km/h.

3. Let x be the cost of one apple pie and y be the cost of one lemon meringue pie. We can set up two equations based on the information given:

6x + 4y = 580

6x + 5y = 94

We can solve for x and y using elimination or substitution. Here is one way to do it using subtraction:

Subtract the second equation from the first equation to eliminate x:

y = 116

Substitute y = 116 into one of the original equations to solve for x:

[tex]6x + 4(116) = 580[/tex]

6x = 16

[tex]x = 8/3 or 2.67[/tex]

Therefore, the cost of one apple pie is S2.67 and the cost of one lemon meringue pie is S116.

4. Let's assume the price of one senior citizen ticket is "S" and the price of one child ticket is "C".

From the given information, we can form two equations:

[tex]3S + 3C = 569[/tex]  ...(1) (sales on the first day)

[tex]5S + 3C = 981[/tex] ...(2) (sales on the second day)

To solve for S and C, we can use any method of solving linear equations (substitution, elimination, or matrix method). Here, we will use the substitution method.

From equation (1), we can express C in terms of S:

C = (569 - 3S)/3

Substituting this value of C in equation (2), we get:

[tex]5S + 3[(569 - 3S)/3] = 981[/tex]

Solving for S:

[tex]5S + 569 - 9S = 2943[/tex]

[tex]-4S = -2374[/tex]

[tex]S = 593.5[/tex]

Therefore, the price of one senior citizen ticket is  [tex]S593.5[/tex]  .

To find the price of one child ticket, we can substitute this value of S in equation (1) and solve for C:

[tex]3(593.5) + 3C = 569[/tex]

[tex]3C = -1518.5[/tex]

[tex]C = -506[/tex]

This doesn't make sense as the price of a ticket cannot be negative. It's possible that there was an error in the given information or in our calculations

5. Let the cost of one package of chocolate chip cookie dough be x, and the cost of one package of gingerbread cookie dough be y.

From the information given in the problem, we can set up the following system of equations:

[tex]8x + 12y = 5364[/tex] (Ming's sales)

[tex]x + 4y = 893[/tex] (Carlos's sales)

To solve for x and y, we can use either substitution or elimination method. Let's use the elimination method by multiplying the second equation by 8 and subtracting it from the first equation:

[tex]8x + 12y = 5364-8x - 32y = -7144-20y = -1780[/tex]

y = 89

Now we can substitute y = 89 into either equation to solve for x. Let's use the second equation:

[tex]x + 4(89) = 893[/tex]

[tex]x = 529[/tex]

Therefore, the cost of one package of chocolate chip cookie dough is $529, and the cost of one package of gingerbread cookie dough is $89.

6. Let the price of a senior citizen ticket be x, and the price of a child ticket be y.

From the information given in the problem, we can set up the following system of equations:

 [tex]3x + 5y = 570[/tex] (first day sales)

[tex]12x + 12y = 2160[/tex] (second day sales)

To solve for x and y, we can use either substitution or elimination method. Let's use the elimination method by multiplying the first equation by 4 and subtracting it from the second equation:

[tex]12x + 12y = 2160[/tex]

[tex]-12x - 20y = -2280[/tex]

[tex]-8y = -120[/tex]

y = 15

Now we can substitute y = 15 into either equation to solve for x. Let's use the first equation:

[tex]3x + 5(15) = 570[/tex]

[tex]3x = 495[/tex]

[tex]x = 165[/tex]

Therefore, the price of a senior citizen ticket is $165, and the price of a child ticket is $15.

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The expression (3*6)+(4*12) is equal to 108.

The expression (3*6)+(4*12) can be written mathematically as 3x6+4x12=108. This can be solved by using the distributive property of multiplication over addition, which states that a*(b+c)=a*b+a*c. This can be applied to the expression in the following way: 3x6+4x12=(3x6)+(4x12)=3x(6+12)+4x(6+12)=3x18+4x18=54+72=108. Therefore, the expression (3*6)+(4*12)=108.

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Answer:

a ≥ 18

p ≤ 275

Step-by-step explanation:

a. The inequality for the minimum age for voting in the United States is:

a ≥ 18

This inequality states that a person's age (represented by 'a') must be greater than or equal to 18 years in order to be eligible to vote in the United States.

b. The inequality for the maximum number of people that can attend a performance in the theater is:

p ≤ 275

This inequality states that the number of people (represented by 'p') attending a performance in the theater must be less than or equal to 275 in order to accommodate all attendees within the seating capacity of the theater.