4 1/2 divided by 3 (fraction problem)

The new price of the television is 15/16 of the original price

Percentages are essentially fractions where the denominator is 100. To show that a number is a percent, we use the percent symbol (%) beside the number.

Represent the original price by x

x × 25/100 = x/4%

The new price will be

x - x/4

= (4x- x)/4

= 3x/4

The new price is now increased by 25%

25/100 × 3x/4

= 1/4 × 3x/4

= 3x/16

the new price

3x/16 + 3x/4

= (12x + 3x)/16

= 15x/16

Therefore the new price is 15/16 of the original price.

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Hence, 1) 6 cm (smallest value), 2) 6 cm, and 3)15.167 cm are the approximate dimensions which will result in a material with the biggest volume (largest value).

A 3D object's surface area is indeed the entire area that all of its faces cover. For instance, the surface area of a cube has been its surface area if we need to determine how much paint is needed to paint it. It's always calculated in square units.

Let the height be y as well as the side length of a square base be x. The rectangular solid's surface area is then determined by:

When we simplify this equation, we obtain:

The rectangular solid's volume is determined by:

V = x² × y = x² × (433.5 - 4x²) / (2x)

When we simplify this equation, we obtain:

V = 216.75x - 2x³

We must determine the value of x that maximizes V in order to determine the dimensions that lead to a solid with the maximum volume. Using V's derivative with x as the base, we can calculate:

dV/dx = 216.75 - 6x²

By setting this to 0 and figuring out x, we obtain:

216.75 - 6x² = 0

x² = 36.125

x ≈ 6.005

When we rewrite the equation for y using this value of x, we obtain:

y ≈ 15.167

Thus, the following dimensions will produce a solid with the largest volume:

1. 6 cm (smallest value)

2. 6 cm

3.15.167 cm (largest value)

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Answer:

Step-by-step explanation:

Using the distance formula:  [tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

    [tex]d=\sqrt{(8-0)^2+(-2-6)^2}[/tex]

        [tex]=\sqrt{64+64} \\[/tex]

        [tex]=\sqrt{128}[/tex]

        [tex]=11.3[/tex]

The solution to the inequality f(x) ≤ 0 is the interval [-6, 2]. In other words, the values of x that satisfy the inequality are those that lie between -6 and 2, inclusive.

To solve the inequality f(x) ≤ 0, we need to find the values of x for which the function f(x) is less than or equal to zero.

We start by factoring the quadratic expression f(x) = x^2 + 4x - 12:

f(x) = (x + 6)(x - 2)

Setting this expression to zero, we get:

(x + 6)(x - 2) = 0

This gives us two solutions: x = -6 and x = 2.

Now, we need to determine the sign of f(x) in the intervals between these two solutions. We can use a sign chart to do this:

x f(x)

-∞ +

-6 0

2 0

+∞ +

From the sign chart, we see that f(x) is positive for x < -6 and for x > 2, and it is negative for -6 < x < 2.

To summarize, the solution to the inequality f(x) ≤ 0 for the function f(x) = x^2 + 4x - 12 is the interval [-6, 2].

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In conclusion for a. If the system is homogeneous , every solution is trivial true. b. If there exists a trivial solution, the system is homogeneous false. c. If there exists a nontrivial solution, there is no trivial solution false.

How to solve?

a. If the system is homogeneous, every solution is trivial.

This statement is true. A homogeneous system of linear equations has the form Ax = 0, where A is the coefficient matrix and x is the vector of variables. The trivial solution is always x = 0, which satisfies the equation. Any other solution would require a nonzero x vector, but then Ax would be nonzero, contradicting the fact that it equals zero in a homogeneous system.

b. If there exists a trivial solution, the system is homogeneous.

This statement is false. A system of linear equations can have a trivial solution (i.e., all variables are equal to zero) without being homogeneous. For example, the system

x + y = 0

2x + 2y = 0

has a trivial solution (x = 0, y = 0) but is not homogeneous.

c. If there exists a nontrivial solution, there is no trivial solution.

This statement is false. A homogeneous system of linear equations can have both trivial and nontrivial solutions. For example, the system

x + y = 0

2x + 2y = 0

has both a trivial solution (x = 0, y = 0) and a nontrivial solution (x = 1, y = -1).

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In light of the cyclist crashes, the likelihood of it being raining is 16/33, or roughly 0.485.

It is predicated on the likelihood that something will occur. The fundamental underpinning of theoretical probability is the justification for probability. For instance, the theoretical chance of receiving a head while tossing a coin is ½. If it rains, let R stand for that, and if the cyclist crashes, let C stand for that. We're looking for P(R|C), or the likelihood that it will rain given the crashes involving cyclists. The Bayes theorem can be used to resolve this issue:

P(R|C) = P(C|R) × P(R) / P(C)

Where P(R) is the likelihood of rain, P(C) is the likelihood that the cyclist will crash regardless of the weather, and P(C|R) is the likelihood that the cyclist will crash given that it is raining.

According to the problem statement, we know that:

P(C|R) = 4/5 (the probability of crashing given that it's raining)

P(C|not R) = 1/10 (the probability of crashing given that it's not raining)

P(R) = 1/4 (the probability of rain)

We may use the rule of total probability to determine P(C):

P(C) = P(C|R) × P(R) + P(C|not R) × P(not R)

where P(not R) = 1 - P(R) = 3/4.

When we change the values, we obtain:

P(C) = (4/5) × (1/4) + (1/10) × (3/4) = 11/40

We can now enter each value into Bayes' theorem as follows:

P(R|C) = (4/5) × (1/4) / (11/40) = 16/33

As a result, there is a 16/33, or roughly 0.485, chance that it had been raining when the cyclists crashed.

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Therefore, the volume of the prism is 200 cubic centimeters.

Volume is a measurement of the amount of space occupied by a three-dimensional object. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet. The volume of an object can be calculated using a variety of formulas, depending on its shape. The concept of volume is used in various fields, including physics, engineering, architecture, and manufacturing.

Here,

Identify the base of the prism. In this case, the base is a right triangle with legs of 5 cm and 8 cm, and a hypotenuse of 11 cm.

Use the formula for the area of a triangle to find the area of the base. The formula for the area of a right triangle is A = (1/2)bh, where b and h are the lengths of the base and height, respectively. In this case, we can use the legs of the triangle as the base and height, since they are perpendicular. So, the area of the base is:

A = (1/2)(5 cm)(8 cm) = 20 cm²

Multiply the area of the base by the height of the prism to find the volume. The height of the prism is given as 10 cm, so:

Volume = (Area of base) x (Height)

= (20 cm²) x (10 cm)

= 200 cm³

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Answer:y=115

Step-by-step explanation:

find the sum of the angles:

=(n-2)180

=(5-2)180

=540

find the value of y:

y=540-(135+112+88+90)

y=115

Answer:

115°

Step-by-step explanation:

the sum of the interior angles of a pentagon is 540°

135+90+112+88+y=540

y= 540-135-90-112-88

y= 155°

Answer:

Area of rectangle, [tex]f(x) = 2x^2 - 2x - 4[/tex].

Step-by-step explanation:

We are given with side lengths of a rectangle are (2x-4) units and (x+1) units. It is required to find the area of rectangle.

The area of a rectangle is equal to the product of its length and breadth. It is given by :

[tex]A=L\times B[/tex]

Let us consider, L = (2x-4) units and B = (x+1) units

Plugging the side lengths in above formula:

[tex]A=(2x-4)\times(x+1)[/tex]

[tex]A = 2x^2 + 2x-4x - 4[/tex]

[tex]A=2x^2-2x-4[/tex]

So, the function that models the area of a rectangle is [tex]f(x) = 2x^2 - 2x - 4[/tex].

As per the equations, the two dogs are either 11 years old and 4 years old, or 4 years old and 11 years old.

Factoring is the process of breaking down a mathematical expression or number into its component parts, which can then be multiplied together to give the original expression or number. In algebra, factoring is often used to simplify or solve equations.

An equation is a statement that shows the equality of two expressions, typically separated by an equals sign (=). The expressions on both sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation.

In the given question,

Let's call the age of the first dog "x" and the age of the second dog "y". We know that their ages add up to 15, so we can write:

x + y = 15

We also know that their ages multiply to 44, so we can write:

x * y = 44

Now we have two equations with two unknowns, which we can solve simultaneously to find the values of x and y.

One way to do this is to use substitution. From the first equation, we can solve for one variable in terms of the other:

y = 15 - x

We can substitute this expression for y into the second equation:

x × (15 - x) = 44

Expanding the left side, we get:

15x - x^2 = 44

Rearranging and simplifying, we get a quadratic equation:

x^2 - 15x + 44 = 0

We can factor this equation as:

(x - 11)(x - 4) = 0

Using the zero product property, we know that this equation is true when either (x - 11) = 0 or (x - 4) = 0.

Therefore, the possible values of x are x = 11 and x = 4.If x = 11, then y = 15 - x = 4, which means that the ages of the two dogs are 11 and 4.

If x = 4, then y = 15 - x = 11, which means that the ages of the two dogs are 4 and 11.

Therefore, the two dogs are either 11 years old and 4 years old, or 4 years old and 11 years old.

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Let a sample space be partitioned into three mutually exclusive and exhaustive events, B1, B2, and, B3.We have to complete the following probability table, prior probabilities, conditional probabilities, joint probabilities, and posterior probabilities, and also we have to round our answers to 2 decimal places.

Given information:Probability of B1, P(B1) = 0.11 Probability of A given B1, P(A | B1) = 0.45 Probability of A intersection B1, P(A ∩ B1) = ?

Probability of B1 given A, P(B1 | A) = ?     Probability of B2, P(B2) = ?     Probability of A given B2, P(A | B2) = 0.62, Probability of A intersection B2,  P(A ∩ B2) = ?     Probability of B2 given A, P(B2 | A) = ?

Probability of B3, P(B3) = 0.38.    Probability of A given B3, P(A | B3) = 0.85.   Probability of A intersection B3,

P(A ∩ B3) = ?

Probability of B3 given A, P(B3 | A) = ?

Total probability of A, P(A) = ?

Total probability of sample space = 1.     Let's complete the probability table:Prior probabilities, Conditional probabilities, Joint probabilities, Posterior probabilities      P(B1) = 0.11P(A | B1) = 0.45P(A ∩ B1) = P(A | B1) * P(B1) / P(A)P(B1 | A) = P(A | B1) * P(B1) / P(A)P(B2) = 1 - (P(B1) + P(B3)) = 1 - (0.11 + 0.38) = 0.51P(A | B2) = 0.62P(A ∩ B2) = P(A | B2) * P(B2) / P(A)P(B2 | A) = P(A | B2) * P(B2) / P(A)P(B3) = 0.38P(A | B3) = 0.85P(A ∩ B3) = P(A | B3) * P(B3) / P(A)P(B3 | A) = P(A | B3) * P(B3) / P(A)Total = 1P(A) = P(A ∩ B1) + P(A ∩ B2) + P(A ∩ B3) = 0.11 * 0.45 + 0.51 * 0.62 + 0.38 * 0.85 = 0.6579.  So, the probability table is as follows:Prior probabilities,Conditional probabilities,Joint probabilities,Posterior probabilities

P(B1) = 0.11P(A | B1) = 0.45P(A ∩ B1) = 0.0495P(B1 | A) = 0.3419P(B2) = 0.51P(A | B2) = 0.62P(A ∩ B2) = 0.3162P(B2 | A) = 0.4857P(B3) = 0.38P(A | B3) = 0.85P(A ∩ B3) = 0.3217P(B3 | A) = 0.1724Total = 1P(A) = 0.6579

Hence, the completed probability table is as shown above.

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The given parametric equations are X(t) = -3/2 + 2t and y(t) = vt² + 16, the rate of change of y with respect to "t" when t = 3 is 6v

We have the parametric equations that are X(t) = -3/2 + 2t and y(t) = vt² + 16.

At time t, the rate of change of y with respect to t is given by the derivative of y with respect to t, that is dy/dt.

So, y(t) = vt² + 16

Differentiating with respect to t, we get

⇒ dy/dt = 2vt.

Now, t = 3 gives us,

y(3) = v(3)² + 16 ⇒ 9v + 16.

Therefore, the rate of change of y with respect to t at t = 3 is

dy/dt ⇒ 2vt ⇒ 2v(3) ⇒ 6v.

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At least 68% of the data falls between 62 and 78, and at least 95% of the data falls between 54 and 86.

To answer this question, we can use the empirical rule, also known as the 68-95-99.7 rule, which tells us that for a normal distribution:

Using this rule, we can estimate the fraction of the data that falls between the following pairs:

Between 54 and 86:

To find the range of values that is within two standard deviations of the mean, we can subtract and add two standard deviations from the mean:

Lower bound: 70 - 2*8 = 54

Upper bound: 70 + 2*8 = 86

So, about 95% of the data falls between 54 and 86.

Between 62 and 78:

To find the range of values that is within one standard deviation of the mean, we can subtract and add one standard deviation from the mean:

Lower bound: 70 - 8 = 62

Upper bound: 70 + 8 = 78

So, about 68% of the data falls between 62 and 78.

Therefore, we can conclude that at least 68% of the data falls between 62 and 78, and at least 95% of the data falls between 54 and 86.

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The given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] is obtained by translating the parent cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] 2 units to the left and 5 units down.

A cubic function is a function of the form that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers.

Cubic function:

[tex]$f(x) = (x+2)^3 - 5$[/tex]

Graphing the cubic function using transformations:

To graph the given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] using transformations from the parent function [tex]$f(x) = (x+2)^3 - 5$[/tex] we can follow the following steps:

a) Start with the parent function [tex]$y=x^3$[/tex] and mark the points on the graph.

b) Translate the graph horizontally by 2 units to the left to obtain [tex]$y=(x+2)^3$[/tex].

c) Shift the graph vertically down by 5 units to get the final graph [tex]$y=(x+2)^3-5$[/tex]

d) Check the intercepts and end behavior of the function to ensure the graph is correct.

Therefore, the given cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] is obtained by translating the parent cubic function [tex]$f(x) = (x+2)^3 - 5$[/tex] 2 units to the left and 5 units down.

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The Given statement "If a distribution is normal, then it is not possible to randomly select a value that is more than 4 standard deviations from the mean"  is a true statement.

A normal distribution is a bell-shaped curve that represents how data is spread out around an average value. About 99.99% of the data in a normal distribution falls within 4 standard deviations from the average.

This means that it is very rare to find a data point that is more than 4 standard deviations away from the average. Therefore, if a distribution is normal, it is not possible to randomly select a value that is more than 4 standard deviation from the mean.

The answer is true.

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The exponential distribution is a probability distribution that models the time between events in a Poisson process, where events occur randomly and independently at a constant average rate.

It is commonly used in reliability theory, queuing theory, and other fields to model the failure or waiting times of systems.

(a) The mean time to decay for Cobalt-60 is 5.27 years.

(b) The standard deviation of the decay time is 2.6355 years.

(c) The 99th percentile is 13.6825 years.

(d) The mean time of the experiment is 15.8175 years and the standard deviation is 4.86788 years.

Note: The answers are calculated based on the exponential distribution of radioactive decay with a half-life of 5.27 years for Cobalt-60.

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Answer:

n=20

Sum of age of men = 50n

when 2 left,

(50n-55-63)/n-2=50-1

50n-55-63=49n-98

50n-55=49n-35

50n-49n=-35+55

n=20

Answer:

the answer is 3/2

Step-by-step explanation:

Answer:

the answer si 3/2

When x=10, the value of y in the equation 3x+4y=34 is y=1.

To find the value of y when x=10 in the equation 3x+4y=34, we can substitute x=10 into the equation and solve for y:

3x+4y=34

3(10) + 4y = 34

30 + 4y = 34

4y = 34 - 30

4y = 4

y = 1

An equation is a mathematical statement that shows the relationship between two or more variables using symbols and numbers. It is a statement that asserts the equality of two expressions. An equation typically consists of two sides, separated by an equals sign (=). Each side of the equation may contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Solving an equation involves manipulating the expressions on both sides of the equation to isolate the variable and find its value. Equations are used in various branches of mathematics, physics, engineering, economics, and other sciences to model and solve problems. For example, the equation "2x + 5 = 11" has two sides, left-hand side (2x + 5) and right-hand side (11), separated by an equals sign.

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Answer:

the answer is c

Step-by-step explanation:

add 14 9 times and then whatever that number is double it and remove the negative sign

Answer: 1

Step-by-step explanation:

because it not zero and not 4

The best choice for our inductive hypothesis is option e. Suppose that IP(S)| = 2" for all sets S with cardinality n, n=0,1,...,k. This states that for all sets S with a cardinality of n, where n is any natural number (from 0 to infinity), the power set of S, P(S), has a cardinality of 2".

Suppose that S is a set. Then the power set of S, P(S) is the set containing all subsets of S.P(∅) = {∅}P({a,b})= {∅, {a}, {b}, {a,b}} Remember that the cardinality of a set is the number of elements in it, i.e. |S|. If S is any finite set, even the empty set, it is the case that |P(S)| = 2". Suppose we want to prove this using induction. Also suppose that this is our base case: Base: n=0. Then S must be the empty set, and IP(∅)| = 1 which is equal to 20. therefore with all these conditions, the answer is option e. IP(S)| = 2" for all sets S with cardinality n, n=0,1,...,kf.

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The probability of randomly drawing a penny is 6/24 or 1/4, since there are 6 pennies out of a total of 24 coins.

How to solve and What is Probability?

The probability of randomly drawing a quarter is 10/24 or 5/12, since there are 10 quarters out of a total of 24 coins. The probability of randomly drawing a coin that is not a penny is 18/24 or 3/4, since there are 18 coins that are not pennies out of a total of 24 coins.

Probability is the branch of mathematics that deals with measuring the likelihood or chance of an event or outcome occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

Probability theory is used to make predictions and informed decisions based on available data in various fields, including statistics, finance, engineering, and science.

It involves understanding and analyzing random events, and determining the likelihood of specific outcomes. Probability is an essential tool for decision-making in various applications, such as risk analysis, game theory, and quality control.

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Answer:

The charity drive was held over two days. On the first day, the 20 artists worked for 6 hours. On the second day, 25 artists turned up and worked for 12 hours. Find the total number of art installations completed over the two days.​

Step-by-step explanation:

Let's start by finding the total number of art installations completed on the first day by the 20 artists who worked for 6 hours. If each artist creates one art installation, then the total number of installations created by the 20 artists is:

20 artists x 6 hours/artist = 120 art installations

On the second day, 25 artists worked for 12 hours. Using the same assumption that each artist creates one art installation, the total number of installations created by the 25 artists is:

25 artists x 12 hours/artist = 300 art installations

Therefore, the total number of art installations completed over the two days is:

120 + 300 = 420

So, the charity drive produced 420 art installations over the two days.

The results showed that the mean difference in assessed value between the two pawn shops was $20.

Value is the worth or importance that something has to an individual or group. It is often determined by ideas such as usefulness, importance, or desirability. Value can also be determined by material or monetary worth, such as the amount of money someone is willing to pay for an item or service.

The research team conducted a hypothesis test to determine if the difference in the assessed values was statistically significant. The null hypothesis was that there was no difference in the assessed values between the two pawn shops. The alternative hypothesis was that there was a statistically significant difference in the assessed values between the two pawn shops. The researchers used a two-tailed t-test to determine if the difference in the assessed values was statistically significant.

The t-test results showed that the difference in assessed values between the two pawn shops was statistically significant, with a p-value of 0.0001. This indicated that the difference in assessed values between the two pawn shops was not due to chance. The research team concluded that the two pawn shops were not assessing jewelry of the same value.

The findings of this research are important for the jewelry industry. This research showed that two pawn shops were assessing jewelry differently, indicating that there is a need for a standardized approach to assess jewelry value. The findings of this study can help improve the accuracy of jewelry valuations and help ensure that consumers are paying a fair price for their jewelry. It can also help jewelers, pawn shops, and other jewelry buyers and sellers determine a fair value for their jewelry purchases.

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First before two trains were [tex]990[/tex] kilometers apart, it will require [tex]5.5[/tex] hours.

Train speed is calculated as total distance traveled divided by travel time. The time it takes for two trains to pass each other is equal to (a+b) / (x+y) if the lengths of the trains, say a or b, are known and they are going at speeds of y and x, respectively.

Typically, a locomotive fueled by electricity or diesel powers trains. If there are several route networks, complicated signaling methods are used. One of the quickest forms of land transportation is rail.

[tex]distance = rate * time[/tex]

distance between trains [tex]= (85 km/h) * t + (95 km/h) * t[/tex]

distance between trains [tex]= (85 + 95) km/h * t[/tex]

distance between trains [tex]= 180 km/h * t[/tex]

Now, we can set up an equation to solve for the time it takes for the trains to be [tex]990[/tex] km apart:

[tex]180 km/h * t = 990 km[/tex]

[tex]t = 990 km / 180 km/h[/tex]

[tex]t = 5.5[/tex] hours

Therefore, it will take [tex]5.5[/tex] hours before the two trains are [tex]990[/tex] km apart.

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The maximum number of coupons that we use for hotel C is 21.  In order to make an equation for this problem, we follow the process given below.

To maximize the number of coupons for hotel C, we should use as many coupons as possible for hotel A and hotel B. Let y be the number of nights we spend at hotel A, and z be the number of nights we spend at the hotel B.Then we have the following equations:

y + z + x = 45 (the total number of nights we stay in hotels A, B, and C)

y <= 25 (we use a maximum of 25 coupons for hotel A)

z <= 20 (we use a maximum of 20 coupons for hotel B)

x <= minimum of (y, z) (we can spend at most one night at hotel C before switching to another hotel)

To maximize x, we need to minimize y and z. Since we cannot spend two consecutive nights at the same hotel, we should alternate between hotels A, B, and C. Thus, we can either start with hotel A or hotel B and then alternate between the two.

Let's assume we start with hotel A. Then we can spend one night at hotel A, one night at hotel C, one night at hotel B, and repeat. This means that y = z = 12 (we spend 12 nights at each of hotel A and hotel B) and x = 21 (we spend 21 nights at hotel C).

Therefore, the maximum number of coupons for hotel C that can be used is 21.

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The probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028,  probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95 and the probability of at least one sprinkler operating correctly in a fire is at least 0.98, we need to have at least 7 sprinklers in the ballroom.

The sprinklers activate correctly or not independently can be found by:

(A) If the probability if all the sprinklers will operate correctly in fire is:

P(all 10 sprinklers operate correctly) = (0.7)¹⁰= 0.028247 (rounded to 3 decimal places)

Therefore, the probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028.

(b) The probability that at least 7 of the sprinklers will operate correctly can be calculated using the binomial distribution with n = 10 and p = 0.7:

P(at least 7 sprinklers operate correctly) = P(X >= 7) = 1 - P(X < 7)

where X is the number of sprinklers that operate correctly.

Using a binomial calculator or table, we can find that:

P(X < 7) = 0.0518 (rounded to 4 decimal places)

Therefore:

P(at least 7 sprinklers operate correctly) = 1 - P(X < 7) = 1 - 0.0518 = 0.9482 (rounded to 2 decimal places)

So, the probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95.

(c) We need to find the minimum number of sprinklers needed so that the probability that at least one operates correctly is at least 0.98. This is equivalent to finding the smallest value of n such that:

P(at least one sprinkler operates correctly) >= 0.98

Using the complement rule, we can rewrite this as:

P(no sprinkler operates correctly) <= 1 - 0.98 = 0.02

The probability that no sprinkler operates correctly in a fire is:

P(no sprinkler operates correctly) = (0.3)¹⁰

So we need to solve for n in the inequality:

(0.3)ⁿ <= 0.02

Taking the natural logarithm of both sides and using the logarithmic rule for inequalities, we get:

n >=ln(0.02) / ln(0.3) ≈ 6.52

Therefore, we need at least 7 sprinklers to ensure that the probability that at least one operates correctly is at least 0.98.

Therefore, The probability that all 10 sprinklers will operate correctly in a fire is approximately 0.028,  probability that at least 7 of the sprinklers will operate correctly in a fire is approximately 0.95 and the probability of at least one sprinkler operating correctly in a fire is at least 0.98, we need to have at least 7 sprinklers in the ballroom.

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The solutions of the system of equations are option (A) (3, 7) and (-3, -5)

To solve the system of equations, use the concept of substitution, which involves solving one equation for one variable and then substituting that expression into the other equation to eliminate one variable and solve for the other variable.

In this case, solve equation (1) for y in terms of x and substitute that expression into equation (2), which allowed us to solve for x. Then we used the values of x to find the corresponding values of y.

Here we have

y = 2x + 1 --- (1)

y = x²+ 2x -8 --- (2)

From (1) and (2)

=>  x²+ 2x - 8 = 2x + 1

Subtract 2x + 1 from both sides

=>  x²+ 2x - 8 - 2x - 1 =  2x + 1 - 2x - 1  

=> x² - 9 = 0

Now add 9 on both sides

=> x² - 9 = 0 + 9

=> x² = 9

=> x = √9  

=> x = ± 3

From (1)

At x = 3

=> y = 2(3) + 1 = 7

At x = - 3

=> y = 2(-3) + 1 = - 5

Therefore,

The solutions of the system of equations are option (A) (3, 7) and (-3, -5)

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Answer:

3c + 19

Step-by-step explanation:

Perimeter: P = a + b + c

P = (c + 10) + (c + 6) + (c + 3) = 3c + 19