Complete the square to re-write the quadratic function in vertex form.
y=x²+7x+3

The volume of the cone is approximately 2941.6 cubic inches.

To begin with, let's recall the formula for the volume of a cone. The volume of a cone is given by the formula V = (1/3)πr²h, where V is the volume, r is the radius of the base, and h is the height of the cone.

We can see that the radius of the base of the cone is 15 inches. However, we do not know the height of the cone yet. To find the height, we can use the fact that the sphere is inscribed in the cone.

We have a right triangle with the height of the cone, the radius of the base of the cone (which is one leg of the triangle), and the diameter of the sphere (which is the hypotenuse of the triangle). Using the Pythagorean theorem, we get:

h² + 15² = 20²

Simplifying this equation, we get:

h² + 225 = 400

h² = 175

h ≈ 13.23

Now that we know the height of the cone, we can plug in the values of the radius and the height in the formula for the volume of the cone. We get:

V = (1/3)π(15²)(13.23) ≈ 2941.6 cubic inches

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If the probability of failure is 0.001, then the probability that

(a) lot is accepted is 0.9277,

(b) lot is rejected on 20th test is 0.000981,

(c) lot is rejected in 10 or fewer trials is 0.01.

Let the random-variable "X" represent number of batteries to test to get the first-battery which fails; in other words, to get first success.

Since, the trials are independent, X has a geometric distribution

Part (a) :

Let X =  the number of trials,

So, the probability that the lot is accepted is :

⇒ P(X=75) = (0.999)⁷⁵×(0.001)⁰ = 0.9277,

Part (b) :

let Y = the numbers of trials before the first failure (geometric distribution and

So, the probability that lot is rejected on 20th-test is :

⇒ P(Y=20) = (0.001)⁰×(0.999)¹⁹ = 0.000981,

Part (c) :

The probability that it is rejected in 10 or fewer trials is :

⇒ 1- P(no Failures)

⇒ 1 – (0.001)⁰(0.999)¹⁰ = 0.01.

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Answer:

Rounding to the nearest centimeter, we get p ≈ 57 cm.

Step-by-step explanation:

We are given a triangle ΔPQR with r = 38 cm, m∠P = 49°, and m∠Q = 127°. We are asked to find the length of side p.

First, let's find m∠R:

m∠R = 180° - (m∠P + m∠Q)

m∠R = 180° - (49° + 127°)

m∠R = 180° - 176°

m∠R = 4°

Now, we have all three angles of the triangle: P = 49°, Q = 127°, and R = 4°. We can use the Law of Sines to find the length of side p:

p/sin(P) = r/sin(R)

Let's plug in the known values:

p/sin(49°) = 38/sin(4°)

Now, solve for p:

p = (sin(49°) * 38) / sin(4°)

p ≈ 56.96 cm

Rounding to the nearest centimeter, we get p ≈ 57 cm.

The probability that the marble drawn from the bag will be either white or black is 2/3 or approximately 0.667.

To calculate this probability, we first need to find the total number of marbles in the bag, which is 10 + 20 + 5 = 35. Then, we need to find the total number of marbles that are either white or black, which is 10 + 20 = 30.

Finally, we divide the number of white or black marbles by the total number of marbles to get the probability:

30/35 = 2/3 ≈ 0.667

Therefore, the probability is approximately 0.667 or 2/3. This means that there is a higher chance of drawing a white or black marble than a grey marble, as there are more white and black marbles in the bag than grey marbles.

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Using the formula, we now know that the surface area of the rectangular prism is 22 in².

A rectangular prism is a polyhedron in geometry that has two parallel and congruent bases.

It also goes by the name cuboid.

Six faces, each with a rectangle shape and twelve edges, make up a rectangular prism.

Its length, width, and height are its three dimensions.

Rectangular tissue boxes, school notebooks, laptops, fish tanks, and big buildings like freight containers, rooms, storage sheds, etc. are a few instances of rectangular prisms in daily life.

So, the formula for the surface area of the rectangular prism:

A = 2(wl+hl+hw)

Now, insert values as follows:

A = 2(wl+hl+hw)

A = 2·(1·1+5·1+5·1)

A = 22 in²

Therefore, using the formula, we now know that the surface area of the rectangular prism is 22 in².

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The equation y=x²+2 (letter C) describes the given function.

In math, there are different functions: linear, quadratic, cubic, exponential and others. Each type of function presents its characteristics. See below:

A linear function can be represented by the standard form for the linear equation is: y= mx+b , for example, y=5x+6. Where: m= the slope and b= the constant term that represents the y-intercept.

Since, the quadratic function can represent a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function, the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.

For answering that the question asks you should test the values of x that represent the value y in the given table.

For x=-2 -> y=-3*(-2)=6 . Then, (-2,6) is a point of the function.

For x=5 -> y=-3*(5)=-15.   Then, (5,27) is not a point of the function.

Thus, the equation y=-3x does not describe the function.

For x=-2 -> y=5*(-2)+16=-10+16=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=5*(5)+16=-25+16=41 .   Then, (5,27) is not a point of the function.

Thus, the equation y=5x+16 does not describe the function.

For x=-2 -> y=(-2)²+2=4+2=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=(5)²+2=25+2=27.   Then, (5,27) is a point of the function.

For x=4 ->  y=(4)²+2=16+2=18.   Then, (4,18) is a point of the function.

For x=7 ->  y=(7)²+2=49+2=51.   Then, (7,51) is a point of the function.

Thus, the equation y=x²+2 describes the function.

For x=-2 -> y=2*(-2)²-2=2*4-2=8-2=6 . Then, (-2,6) is a point of the function.

For x=5 ->  y=2*(5)²-2=2*25-2=50-2=48.   Then, (5,27) is not a point of the function.

Thus, the equation y=2x²-2 does not describe the function.

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Answer:

There were 24 chocolates in the box to begin with.

Step-by-step explanation:

Let's use working backwards to solve this problem.

We know that there are 12 chocolates left in the box. Let's call the number of chocolates that were in the box to begin with "x".

After eating 1/4 of what was left, Forest had 3/4 of the chocolates left. So, we can set up an equation:

3/4 (x - 1/3x) = 12

We can simplify this equation by multiplying both sides by 4/3:

x - 1/3x = 16

Multiplying both sides by 3:

2x = 48

x = 24

Therefore, there were 24 chocolates in the box to begin with.

The probability that a standard normal variable exceeds 2 is approximately 0.0228

To solve this problem, we need to use the central limit theorem, which states that the sum of a large number of independent and identically distributed random variables tends to follow a normal distribution, regardless of the distribution of the individual variables.

In this case, we have 25 independent individuals, each with a normal distribution of weights with mean 160 pounds and standard deviation 30 pounds. The total weight of the 25 individuals is the sum of these 25 random variables.

The mean weight of a single person is 160 pounds, so the mean weight of 25 people is 25 times 160, which is 4000 pounds. The standard deviation of the sum of 25 random variables is the square root of the sum of the variances of the individual variables, which is the square root of 25 times 30 squared, or 150 pounds.

The probability that the total weight of the 25 people exceeds the design limit of 4300 pounds can be calculated using the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. We can convert the total weight of the 25 people to a z-score using the formula

z = (x - mu) / sigma

where x is the total weight, mu is the mean weight, and sigma is the standard deviation of the sum of the individual weights.

z = (4300 - 4000) / 150 = 2

The probability that a standard normal variable exceeds 2 is approximately 0.0228.

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The area of a square is given by the formula A = s^2, where s is the length of a side of the square. If the area of the square is 100 sq meters, then:

s^2 = 100

Taking the square root of both sides, we get:

s = 10

Therefore, the length of a side of the square is 10 meters. Since all four sides of a square are equal, the perimeter of the square is:

4s = 4(10) = 40 meters

we conclude that △ACO is congruent to △BCO by the side-angle-side (SAS) congruence theorem.

We are given that angle AOB is a central angle of circle O and that angle ACB is a circumscribed angle of circle O. We see that AO ≅ BO because they are radii of the same circle.

We also know that AC ≅ BC since they are both tangent to the same circle and intersect at point C.

Thus, angle ACB and angle AOB have the same measure, which means that triangle ACO and triangle BCO are isosceles with AC ≅ BC and ∠ACO ≅ ∠BCO.

Using the reflexive property, we see that angle COA is congruent to angle COB since they are both vertical angles.

Therefore, we conclude that △ACO is congruent to △BCO by the side-angle-side (SAS) congruence theorem.

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The total number of campers surveyed is 150.

The symbol for percentage is "%". Percentages are commonly used in a wide range of contexts, such as in finance, statistics, and everyday life.

We can use the information provided about the percentage of campers in each time interval to set up an equation to solve for the total number of campers surveyed.

Let n be the total number of campers surveyed. Then:

15% of n spend more than 3 hours reading daily.

10% of n spend between 2 and 3 hours reading daily.

35% of n spend less than 1 hour reading daily.

40% of n spend between 1 and 2 hours reading daily.

We know that 60 campers spend between 1 and 2 hours reading daily, which represents 40% of the total number of campers surveyed. So we can set up the following equation:

0.4n = 60

Solving for n, we get:

n = 60 / 0.4 = 150

Therefore, the total number of campers surveyed is 150.

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Answer: D

Step-by-step explanation:
To narrow it out, Histograms dont have spaces meaning this chart wouldnt work as a histogram, and A and C are just plain incorrect

Answer:

Jayla selected 14 bean bags randomly with replacement from a box of 100 bean bags, and 6 of them were red. To estimate how many red bean bags are in the box, we can use proportion:

Let's call "x" the number of red bean bags in the box. Then we can set up a proportion:

(red bean bags in the box) / (total bean bags in the box) = (red bean bags selected) / (total bean bags selected)

x / 100 = 6 / 14

Simplifying the right side, we get:

x / 100 = 3 / 7

Multiplying both sides by 100, we get:

x = 300 / 7

Rounding to the nearest whole number, we get:

x ≈ 43

Therefore, based on the data, we can estimate that there are approximately 43 red bean bags in the box.

Answer + Explanation:

To estimate how many red bean bags are in the box, we can use a sampling method called "balloting." Balloting is a way of estimating the number of items of a specific type in a larger population by taking a random sample and counting how many of that type are in the sample.

In this case, we have a population of 100 bean bags and a sample size of 14. We can calculate the sample proportion of red bean bags by dividing the number of red bean bags in the sample by the total number of bean bags in the sample, which gives:

(4/14) × 100% ≈ 28.6%

This means that, on average, we would expect 28.6% of the 100 bean bags in the box to be red. To estimate the actual number of red bean bags, we can multiply the sample proportion by the total number of bean bags in the box:

28.6% × 100 = 28.6

So the closest estimate for the number of red bean bags in the box is 29.

Note that this is just an estimate, and the actual number of red bean bags in the box may be higher or lower than the estimated value.

Yes, Lucia will be able to form a triangular-shaped sandbox using the railroad ties without cutting them.

To determine if a triangle can be formed, you can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side.

In this case, the sides are 10 feet, 12 feet, and 8 feet. Let's check if the Triangle Inequality Theorem holds true for each combination:

1. 10 feet + 12 feet = 22 feet > 8 feet (Yes)
2. 10 feet + 8 feet = 18 feet > 12 feet (Yes)
3. 12 feet + 8 feet = 20 feet > 10 feet (Yes)

Since all three combinations satisfy the Triangle Inequality Theorem, Lucia can create a triangular-shaped sandbox with the given railroad ties.

The Triangle Inequality Theorem is a mathematical principle that states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, if a, b, and c are the lengths of the sides of a triangle, then:

a + b > c

b + c > a

a + c > b

If any of these inequalities is not true, then the given lengths do not form a valid triangle.

This theorem is important in geometry and other fields that involve measurements of distance or length. It is also used in many practical applications, such as designing bridges, buildings, and other structures, where the strength and stability of the structure depend on the lengths and angles of the sides of the triangles involved.

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The length of the line can be made is 5x/8 inch, where 'x' is the number of strips.

Length is a measure of the size of an object or distance between two points, usually measured in units such as meters, centimeters, inches, feet, or miles.

In the given problem to find the length of a line we need to multiply the length of each strip by the number of strips

Here we have

The length of the strip = 5/8

The length of the line will depend on the number of strips

Let there be 'x' number of strips

The length of the line can be made = x × 5/8

=> 5x/8

Therefore,

The length of the line can be made is 5x/8 inch, where 'x' is the number of strips.

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A given function or graph is needed to find the average rate of change between an interval

The equations model the possible dimensions of the playground is option (A) x(x + 8) = 256.

A statement that affirms the equality of two expressions connected by the equals symbol "=" is known as an equation. For illustration, 2x - 5 = 13. 2x - 5 and 13 are expressions in this case. These two expressions are joined together by the sign "=".

The area of the rectangular playground is given as 256 square yards. Let x be the length of the shorter side of the playground in yards. Then the length of the longer side can be expressed as x + 8 yards, since it is 8 yards longer than the shorter side.

The area of the rectangle is given by the product of its length and width. Therefore, we can write:

x(x + 8) = 256

This equation represents the possible dimensions of the playground, where x is the length of the shorter side and x + 8 is the length of the longer side.

So, the correct equation that models the possible dimensions of the playground is:

x(x + 8) = 256.

Therefore, the equations model the possible dimensions of the playground is option (A) x(x + 8) = 256.

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Answer:

[tex]2x+5y=20[/tex]

Step-by-step explanation:

Rearranging the equation of the given line into intercept form,

[tex]5y+2x=10 \\ \\ \frac{x}{5}+\frac{y}{2}=1[/tex]

This means the intercepts are [tex](5,0)[/tex] and [tex](0,2)[/tex].

We need to find the equation of the line with intercepts [tex](0,4)[/tex] and [tex](10,0)[/tex]. This equation is [tex]\frac{x}{10}+\frac{y}{4}=1[/tex], which rearranges to give [tex]2x+5y=20[/tex].

The value οf y fοr pοint P οn the unit circle in the third quadrant with x-cοοrdinate -6/7 is -sqrt(13)/7.

PWe knοw that the pοint  is in the third quadrant and is distance 1 away frοm the οrigin, which is at (0,0). We alsο knοw that there is a pοint at (1,0) οn the x-axis, which is tο the right οf the οrigin.

Tο sοlve fοr y, we can use the Pythagοrean theοrem tο set up an equatiοn based οn the distance fοrmula:

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

Squaring bοth sides, we get:

[tex]x^2 + y^2 = 1[/tex]

Since we knοw that x = -6/7, we can substitute that in and sοlve fοr y:

[tex](-6/7)^ + y^2 = 1[/tex]

Simplifying, we get:

[tex]36/49 + y^2 = 1[/tex]

Subtracting 36/49 frοm bοth sides, we get:

y^2 = 13/49

Taking the square rοοt -

οf bοth sides yields:

[tex]y = +/- \sqrt{(13)} /7[/tex]

Because P is lοcated in the third quadrant, y must be negative. Therefοre, the value οf y in simplest fοrm is:

[tex]y = - \sqrt{(13)}/7[/tex]

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Answer:

66 hours.

Step-by-step explanation:

6 cars x 34 hours per car = 204

6 cars x 23 hours per car= 138 hours

now subtract:

204 - 138 = 66

therefore, it took Theo 66 hours longer to wash all the cars last week than last week.

if x = 25, quadrilateral ABCD will be a trapezoid with AB || CD.thus, the answer is (C) 25.

What is trapezoid?

For quadrilateral ABCD to be a trapezoid, it must have a pair of parallel sides. In other words, either AB || CD or AD || BC. We can use the angle measures to determine which condition must be satisfied.

Trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid, and the non-parallel sides are called the legs. The height of a trapezoid is the perpendicular distance between the bases. The area of a trapezoid can be calculated by taking the average of the lengths of the two bases and multiplying by the height. The formula for the area of a trapezoid is: Area = (base1 + base2) / 2 * height

If AB || CD, then opposite angles must be supplementary, which means that:

A + D = 180°

Substituting the given angle measures, we get:

3x + (5x-20) = 180

Simplifying, we get:

8x - 20 = 180

Adding 20 to both sides, we get:

8x = 200

Dividing both sides by 8, we get:

x = 25

Therefore, if x = 25, quadrilateral ABCD will be a trapezoid with AB || CD.

Thus, the answer is (C) 25.

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The temperature at 7:00 a.m. was 67 degrees.

To find the temperature at 7:00 a.m., we need to consider that it was 20 degrees cooler than at 1:00 p.m.
1. Identify the given information: At 1:00 p.m., the temperature was 87 degrees, and it was 20 degrees cooler six hours earlier.
2. We need to find the temperature at 7:00 a.m., which is six hours before 1:00 p.m.
3. Since the temperature was 20 degrees cooler at 7:00 a.m.

we will subtract 20 from the temperature at 1:00 p.m.
4. Perform the subtraction:

87 degrees (1:00 p.m. temperature) - 20 degrees (cooler difference) = 67 degrees.
5. The result, 67 degrees, is the temperature at 7:00 a.m.
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The higher the value of Cronbach's alpha, the higher the internal consistency reliability of the assessment.

A measurement of internal consistency that allows researchers to determine how well the different items/questions within a test measure different aspects of the same topic is known as reliability.

Reliability refers to the consistency of the results of an assessment. In other words, it refers to the extent to which an assessment yields stable and consistent results. Researchers use various methods to assess the reliability of an assessment.

These include test-retest reliability, inter-rater reliability, and internal consistency reliability.

In this case,

The internal consistency reliability is the method that allows researchers to determine how well the different items/questions within a test measure different aspects of the same topic.

Internal consistency reliability is a method that measures the consistency of the results of an assessment by examining the relationships among the different items/questions within the assessment. It looks at the degree to which the items/questions within an assessment measure the same thing or construct.

If the different items/questions within an assessment are measuring the same thing, then the assessment is said to have high internal consistency reliability. If the items/questions are measuring different things, then the assessment is said to have low internal consistency reliability.

Researchers use various statistical methods to assess internal consistency reliability. One of the most common methods is Cronbach's alpha, which measures the degree to which the items/questions within an assessment are correlated with each other.

The higher the value of Cronbach's alpha, the higher the internal consistency reliability of the assessment.

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Answer: y= 4x+2

Step-by-step explanation: meaning of slope - the amount of bacteria multiplies by 4 as each minute passes

(sorry English isn't my first language)

According to the given information, the solutions to the equation 7x²-34=2x²+16 are x = √10 and x = -√10.

What is quadratic equation?

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0 where a, b, and c are constants, and x is the variable.

To solve for x in the equation 7x²-34=2x²+16 using square roots, we can first simplify the equation by moving all the x² terms to one side and all the constant terms to the other side:

7x² - 2x² = 16 + 34

5x² = 50

5x²/5 = 50/5

x² = 10

Finally, we can take the square root of both sides (remembering to include both the positive and negative square root, since x could be positive or negative):

x = ±√10

Therefore, the solutions to the equation 7x²-34=2x²+16 are x = √10 and x = -√10.

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The fraction of the check and tip did John pay is 5/24.

What are rational numbers?

It is possible to express rational numbers in the form pq, where p and q are integers and q0. The distinction between fractions and rational numbers is that the numerator or denominator of a fraction cannot be negative. As a result, the numerator and denominator of a fraction are whole numbers (denominator 0), as opposed to integers in the case of rational numbers.

Here, we have

Given: A group of three friends ate dinner at a restaurant. When they settled the check and tip, Peter paid 4/5 as much as John paid, and John paid 1/3 as much as Ralph paid.

we have to find a fraction of the check and tip did John pay.

p, j, r, the fractions that each paid.

p + j + r = 1....(1)

p = (4/5)j and j = (1/3)r

From equation(1), we get

(4/5)j + j + 3j = 1

j(4/5 + 1 + 3) = 1

(24/5)j = 1

j = 5/24

We have

r = 3j

r = 3(5/24)

r = 5/8

Hence, the fraction of the check and tip did John pay is 5/24.

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Area of trapezoid= 1/2 (13.4+5.6)×4

= 1/2 ×19

=76/2

=38yd

The value of the expression using PEDMAS is 7.

The acronym PEMDAS, which can alternatively stand for "Please Forgive My Dear Aunt Sally," is a mnemonic tool for remembering the order of operations in mathematical and algebraic formulas. Its acronym is:

P - Parenthesis (operations inside parentheses are done first)

E: Exponents (operations involving exponents are done next)

Divide and multiply is abbreviated MD (these operations are done left to right, whichever comes first)

AS stands for addition and subtraction (these operations are done left to right, whichever comes first). No matter how they approach it or how they understand it, PEMDAS helps to ensure that the same statement is assessed in the same way by various people. In mathematics, it is crucial to understand this idea, especially when working with complicated expressions or formulae.

The given expression is 1/2 x (48 ÷ 6) – 2 + 5.

Simplifying the expressions we have:

1/2 x (48 ÷ 6) – 2 + 5

= 1/2 x 8 - 2 + 5

= 4 - 2 + 5

= 7

Hence, the value of the expression using PEDMAS is 7.

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