Forest bought a box of chocolates for himself last week. However, by the time forest got home he had eaten 1/3 of the box. As Forest was putting the groceries away, he at 1/4 of what was left. There are now 12 chocolates left in the box.

How many chocolates were in the box to begin?

Be sure to show and explain all your reasoning.

The amount of work required to stretch the spring from 30.0 cm to 36.0 cm is 1.411 joules, which can be rounded to three decimal places.

According to Hooke's Law, the amount of work required to stretch or compress a spring by a certain amount is given by the formula:

W = (1/2) k (x2 - x1)²

where W is the work done (in joules), k is the spring constant (in newtons per meter), x1 is the initial displacement (in meters), and x2 is the final displacement (in meters).

In this case, the spring has a natural length of 30.0 cm, which is equivalent to 0.3 meters. To find the spring constant, we can use the fact that a 20.0-N force is required to keep it stretched to a length of 42.0 cm, which is equivalent to 0.42 meters.

Using Hooke's Law, we have

F = k (x2 - x1)20.0 N

= k (0.42 - 0.3) m

=> k = 80.0 N/m

Now we can use Hooke's Law again to find the amount of work required to stretch the spring from 30.0 cm to 36.0 cm, which is equivalent to 0.36 meters.

Using Hooke's Law, we have:

F = k (x2 - x1)W

= (1/2) k (x2 - x1)²W

= (1/2) (80.0 N/m) (0.36 - 0.3) m²W

= 1.411 J

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The radius of the spherical boulder is half the diameter, so:

radius = 24 ft / 2 = 12 ft

The formula for the volume of a sphere is:

V = (4/3)πr³

Substituting the given values, we get:

V = (4/3) x 3.14 x (12 ft)³

V = 7238.08 cubic feet

Rounding to the nearest whole number, we get:

V ≈ 7238 cubic feet

Answer: 1.5 feet (rounded to the nearest tenth) or 1.4644

Answer: The general equation for a circle with center (h, k) and radius r is:

(x - h)^2 + (y - k)^2 = r^2

To find the equation of the circle that passes through the points (1, 1), (1, 3), and (9, 2), we can use the fact that the perpendicular bisectors of any two chords of a circle intersect at the center of the circle.

The midpoint and slope of the chord connecting (1, 1) and (1, 3) are:

Midpoint: ((1+1)/2, (1+3)/2) = (1, 2)

Slope: undefined (since x values are the same)

The perpendicular bisector of this chord passes through the midpoint (1, 2) and has a slope of 0. Therefore, its equation is:

y - 2 = 0

Simplifying this equation gives:

y = 2

Similarly, the midpoint and slope of the chord connecting (1, 1) and (9, 2) are:

Midpoint: ((1+9)/2, (1+2)/2) = (5, 1.5)

Slope: (2 - 1)/(9 - 1) = 1/8

The perpendicular bisector of this chord passes through the midpoint (5, 1.5) and has a slope of -8 (the negative reciprocal of the slope of the chord). Therefore, its equation is:

y - 1.5 = -8(x - 5)

Simplifying this equation gives:

y = -8x + 41.5

The intersection of the two perpendicular bisectors (y = 2 and y = -8x + 41.5) is the center of the circle. Solving for x gives:

2 = -8x + 41.5

8x = 39.5

x = 4.9375

Substituting x = 4.9375 into either equation for the perpendicular bisectors gives:

y = 2

Therefore, the center of the circle is (4.9375, 2) and the radius is the distance from the center to any of the three given points. For example, the distance from (4.9375, 2) to (1, 1) is:

r = sqrt((4.9375 - 1)^2 + (2 - 1)^2) = sqrt(24.9219) = 4.992

So the equation of the circle is:

(x - 4.9375)^2 + (y - 2)^2 = (4.992)^2

Expanding and simplifying this equation gives:

x^2 - 9.875x + 24.526 + y^2 - 4y + 0.076 = 24.916

Rearranging terms gives:

x^2 + y^2 - 9.875x - 4y - 0.34 = 0

So the general equation for the circle is:

x^2 + y^2 - 9.875x - 4y - 0.34 = 0

Step-by-step explanation:

To better understand your question, it is important to identify the key terms and concepts that are being used.

Title: Understanding Terms in a Question

1. . Look for specific words or phrases that relate to the topic at hand, and try to understand their meaning within the context of the question.

2. It can also be helpful to break the question down into smaller parts or sub-questions, and focus on each one individually.

3. If you are unsure about the meaning of a particular term, do some research or consult a dictionary or other reference source.

4. By taking the time to understand the terms and concepts in a question, you can improve your ability to answer it accurately and effectively.

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Answer:

parallelogram

Step-by-step explanation:

opposite angles and sides are congruent

The probability that the average size of the fish you caught is more than 14 inches is 0.1635 or 0.164

The size of bass caught in Strawberry Lake is normally distributed with a mean of 12 inches and a standard deviation of 5 inches.

If you catch 6 fish, the probability that the average size of the fish you caught is more than 14 inches is 0.202.

Here,

we need to find the probability that the average size of the fish you caught is more than 14 inches.

Let us denote the size of the bass caught in Strawberry Lake by X.

Then, X ~ N(μ = 12, σ = 5) represents the normal distribution of the size of bass caught in Strawberry Lake.

Let Y be the sample mean of 6 bass caught in the Strawberry Lake.

Then,

We know that Y ~ N(μ = 12, σ = 5/√6) represents the sampling distribution of the sample mean,

where σ = 5/√6

= 2.0412 (approx).

We are given that we have caught 6 fish.

Therefore, the sample size n = 6.

Then,

The probability that the average size of the fish you caught is more than 14 inches can be obtained as follows:

P(Y > 14) = P((Y - μ)/σ > (14 - μ)/σ)

= P(Z > (14 - 12)/2.0412)

= P(Z > 0.977)

= 1 - P(Z < 0.977) (as the standard normal distribution is a continuous distribution)

Using the standard normal distribution table, we get P(Z < 0.977) = 0.8365 (approx) Therefore, P(Y > 14) = 1 - P(Z < 0.977) = 1 - 0.8365 = 0.1635 (approx)

Therefore,

The probability that the average size of the fish you caught is more than 14 inches is 0.1635 or 0.164 (approx).

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The product of twice a number and four is 8 times the number.

The product means a number that you get by multiplying two or more other numbers together.

The product of twice a number and four can be represented algebraically as:

4(2x)

where x is the number we are referring to.

Simplifying the expression, we get:

4(2x) = 8x

Therefore, the product of twice a number and four is 8 times the number.

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In a cube, a total of 12 lines can be formed.

It has 8 vertices and 12 edges. In geometry, a line is a straight geometric figure that extends indefinitely in both directions. The lines in a cube are the straight lines that connect two vertices or edges of the cube. When two points are connected with a line, it creates an edge. In a cube, there are 12 edges. Each edge is shared by two faces, and the edges are the line segments that make up the sides of the cube.

Thus, the number of lines that can be formed in a cube is 12. When two or more lines meet at a point, it is called a vertex. A cube has eight vertices or corners. When a line is drawn between any two vertices of a cube, it is called a diagonal. In a cube, there are 12 diagonals. A diagonal is a straight line segment that connects two non-adjacent vertices of a polygon. So, the number of lines that can be formed in a cube is 12, which includes the 12 edges.

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The expression is undefined, so the sign cannot be determined.

How to determine sign for the given problem?

The following given expression can be more simplified as follows:

-9 * (0 - 3)    -    (9 * (-3/0))

= -9 * (-3) - 9 * undefined [Note: Division by zero is undefined]

= 27 - undefined

As anything subtracted from undefined remains undefined, the overall result is undefined.

Therefore, the sign of the expression cannot be determined.

Undefined values represent the absence of a meaningful result or outcome. In this case, the expression involves division by zero, which is undefined. As a result, any operation involving an undefined value will also be undefined, including subtraction. Thus, the overall result is undefined.

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Answer:

ITS ZEROOOO

Step-by-step explanation:

i guesses and ended up getting it right.

The values of the missing sides a and c are 3√6/2 and 3√2 respectively using the trigonometric ratio of tan 45° for the right triangle.

The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.

The basic trigonometric ratios includes;

sine, cosine and tangent.

using the trigonometric ratio of tan 60°

{recall that cos 45° and sin 45° are equal to √2/2}

cos 45° = 3/c {adjacent/hypotenuse}

c = 3 × 2/√2 {cross multiplication}

c = 3√2

sin 45° = a/3√2 {opposite/hypotenuse}

a = 3√2 × √3/2

a = 3√6/2

Therefore, the values of the missing sides a and c are 3√6/2 and 3√2 respectively using the trigonometric ratio of tan 45° for the right triangle.

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Volume of the sphere is equal to two-third of the volume of a cylinder.

Volume is a mathematical concept that refers to the amount of space occupied by a three-dimensional object. It is expressed in cubic units, such as cubic meters or cubic centimeters, and can be calculated using a specific formula for each type of geometric shape.

A sphere is a perfectly round, three-dimensional object that looks like a ball. It has no edges or vertices and is symmetric around its center point. The formula for calculating the volume of a sphere is V = 4/3 πr³, where V is the volume, π is a mathematical constant approximately equal to 3.14, and r is the radius of the sphere.

True.

Assume that both the sphere's and cylinder's radius is r.

Considering that the base's diameter equals the cylinder's height

⇒h = 2r

Sphere radius = cylinder radius = r

Assuming the conditions are met, the sphere's volume:

=  [tex]\frac{2}{3}[/tex] × Volume of cylinder

= [tex]\frac{4}{3}[/tex]πr³= [tex]\frac{2}{3}[/tex] × πr² × 2r

=[tex]\frac{4}{3}[/tex]πr³= [tex]\frac{4}{3}[/tex]πr³

Hence, volume of the sphere is equal to two-third of the volume of a cylinder.

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The complete question is:

The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.

The equation that gives us the height of point P after the windmill blade rotates by an angle of θ is  h = 1.5 * sin(π/4 + θ).

Here, O is the center of the windmill, P is the point at the tip of the blade, and a is the angle through which the blade has rotated. We are given that the center of the windmill is 6 feet off the ground and the blades are 1.5 feet long.

We know that OP is the length of the blade, which is 1.5 feet. To find the angle a, we need to take into account the starting position of P. We are told that P is currently rotated π/4 radians from the point directly to the right of O. This means that if we rotate the blade by an angle of a, P will end up at an angle of π/4 + a from the point directly to the right of O.

where θ is the angle through which the blade has already rotated.

Putting it all together, we have:

h = 1.5 * sin(π/4 + θ)

This equation gives us the height of point P after the windmill blade rotates by an angle of θ.

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Answer:

The three most common methods for solving quadratic equations are factoring, completing the square, and the quadratic formula.

Here's a quadratic equation you can solve by factoring:

x^2 - 4x + 3 = 0

(x - 1)(x - 3) =0

x = 1, 3

a. the sampling distribution is  [tex]\sqrt{[(0.17 \times (1-0.17)) / n]}[/tex]

b. The probability that the sample proportion is within 0.03 of the population proportion is 0.4101.

c. The probability that the sample proportion is within 0.03 of the population proportion for a sample of 200 households is 0.7738.

a. The sampling distribution of the sample proportion, [tex]\bar p[/tex], can be approximated by a normal distribution with mean equal to the population proportion, p, and standard deviation equal to the square root of [tex][(p \times (1-p)) / n][/tex],

where n is the sample size.

Given that p = 0.17 and assuming a large enough sample size, we can use the formula to calculate the standard deviation of the sampling distribution:

Standard deviation = [tex]\sqrt{[(0.17 \times (1-0.17)) / n]}[/tex]

b. To find the probability that the sample proportion will be within a certain range of the population proportion, we need to calculate the z-score for the lower and upper bounds of that range and then find the area under the normal curve between those z-scores.

Let's say we want to find the probability that the sample proportion is within 0.03 of the population proportion. This means we want to find

P([tex]\bar p[/tex]- p ≤ 0.03) = P(([tex]\bar p[/tex]-- p) / [tex]\sqrt{[(p \times (1-p)) / n]}[/tex] ≤ 0.03 / [tex]\sqrt{[(p \times (1-p)) / n][/tex]

We can use the standard normal distribution and z-scores to find this probability:

[tex]z_1[/tex] = (0.03 / [tex]\sqrt{(0.17 \times (1-0.17)/n}[/tex])

[tex]z_2[/tex] = (-0.03 / [tex]\sqrt{(0.17 \times (1-0.17))/n}[/tex])

We can find the probability that the z-score is between [tex]z_1[/tex] and [tex]z_2[/tex]:

P([tex]z_1[/tex]≤ Z ≤ [tex]z_2[/tex]) = P(-0.541 ≤ Z ≤ 0.541) = 0.4101

c. To answer part (c), we need to specify the sample size. Let's say we are taking a sample of 200 households.

Using the formula for standard deviation of the sampling distribution from part (a), we get:

Standard deviation = [tex]\sqrt{(0.17 \times(1-0.17)) / 200}[/tex] = 0.034

Now we can repeat the same steps as in part (b) with this standard deviation:

[tex]z_1[/tex] = (0.03 / 0.034)

[tex]z_2[/tex] = (-0.03 / 0.034)

P([tex]z_1[/tex]≤ Z ≤ [tex]z_2[/tex]) = P(-0.882 ≤ Z ≤ 0.882) = 0.7738

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Answer:

452.39

Step-by-step explanation:

Used volume formula

Answer:

V = 452.4

Step-by-step explanation:

V = πr²h

(3.14)(6)²(4) = 452.38934211

On the basis of similarity of triangles, the triangles in Q9 are similar by AA similarity whereas in Q10 the triangles are similar by SAS similarity.

What is similarity?

Any Two triangles are said to be same or similar if they have the same ratio of the corresponding sides in the triangles and equal pair of corresponding angles in the triangles. If two or more figures or polygons having the same shapes, but their sizes are not same, then those figures or polygons are called similar. To be congruent, the shapes or polygons must have equal angles and equal sides.

Q9)Consider the ΔTUS and ΔUVW

∠STU=∠UVW=(72°)

∠TUS=∠VUW {vertically opposite angles}

The given triangles are similar by AA criteria.

Q10)Consider the ΔKDH and ΔBAD

∠H = ∠A = 38 °

[tex]\frac{KH}{BA} =\frac{DH}{AD}[/tex]

[tex]\frac{20}{28} = \frac{15}{21} = \frac{5}{7}[/tex]

The above triangles are similar by SAS

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Answer:

t=0,  t=4/3

Step-by-step explanation:

can anyone answer my math questions

The equivalent expressions of the given expressions are value: ∛-48a⁶b⁹ = ∛- (2)³(6) (a)²b³ = 2a²b³∛-6. [2] √48a⁶b⁹ = √(4)²(3) (a)³b⁴b = 4|a³|b⁴ √3b [3] ∛-48a⁷b⁸ = -2a³b² ∛6ab².

A mathematical expression is a sentence without the equal sign that can include variables, operators, and integers. It can be assessed or simplified, but there isn't a single, perfect answer. An expression is something like 3x + 2y.

On the other hand, an equation is a mathematical statement that uses the equal sign to denote the equivalence of two expressions. You can solve equations to determine the values of the variables that fulfil the equality. An equation is, for instance, 3x + 2y = 7.

The given expressions can be simplified by taking the factors as follows:

∛-48a⁶b⁹ = ∛- (2)³(6) (a)²b³ = 2a²b³∛-6.

√48a⁶b⁹ = √(4)²(3) (a)³b⁴b = 4|a³|b⁴ √3b

∛-48a⁷b⁸ = -2a³b² ∛6ab²

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The probability of exactly three successes in eight trials is approximately 26.61%

To find the probability of exactly three successes in eight trials of a binomial experiment with a success probability of 45%, use the binomial probability formula:

P(x) = C(n, x) * [tex]p^x[/tex] * [tex](1-p)^{(n-x)}[/tex]

where n is the number of trials, x is the number of successes, p is the probability of success, and C(n, x) is the combination function representing the number of ways to choose x successes from n trials.

In this case, n = 8, x = 3, and p = 0.45.

Calculate the probability:

P(3) = C(8, 3) * [tex]0.45^3[/tex] * [tex](1-0.45)^{(8-3)}[/tex]

P(3) = 56 * 0.091125 * 0.1721865

P(3) ≈ 0.2661

So, the probability of exactly three successes in eight trials is approximately 26.61%.

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By trigonometry Taking a look at the equation's left side 1-COSX/SINX is = (SINX/SINX) - (COSX/SINX) = (SINX - COSX)/SINX.

Several fields, including architecture, physics, astronomy, surveying, oceanography, navigation, electronics, and many more, use trigonometry. These are a few instances:

- In physics, trigonometry is used to calculate forces and analyze waves. - In astronomy, trigonometry is used to calculate distances between stars and planets. - In surveying, trigonometry is used to calculate distances and heights of objects. - In oceanography, trigonometry is used to calculate the heights of waves and tides in oceans.

We must make one side of the equation appear to be the other side in order to prove that 1-COSX/SINX=SINX/1+COSX.

Taking a look at the equation's left side first:

1-COSX/SINX is = (SINX/SINX) - (COSX/SINX) = (SINX - COSX)/SINX.

Let's now adjust the equation's right side:

SINX/1+COSX = (SINX/1+COSX) * (1-COSX/1-COSX)

= (SINX - SINXCOSX)/(1-COSXSINX)

= (SINX - COSXSINX)/(1-COSXSINX)

= (SINX - COSX)/((1-SINXCOSX)/( (1-SINXCOSX)

As the identities on both sides are equal to (SINX - COSX)/SINX, we can conclude that they are correct.

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The two expressions to show how much money Melissa would have after 10 weeks are:
1. 900 - 250 - 10x
2. 900 - (10 × 25 + 10x)

Expression 1: To calculate how much money Melissa would have after 10 weeks, first find the total amount she spends

on gas in 10 weeks by multiplying the weekly gas cost by the number of weeks:

25  10 = 250.

Next, find the total amount she spends on clothes in 10 weeks by multiplying the weekly clothes cost (x) by the number of weeks:

x × 10 = 10x.

Now, subtract both of these expenses from her initial savings to find the remaining balance:

900 - 250 - 10x.

Expression 2: You can also find the total expenses in 10 weeks by adding the gas and clothes costs together:

25 + x.

Then, multiply this combined weekly cost by 10 to find the total cost in 10 weeks:

10(25 + x) = 10 × 25 + 10x.

Finally, subtract the total cost from her initial savings:

900 - (10 × 25 + 10x).

So, the two expressions to show how much money Melissa would have after 10 weeks are:

1. 900 - 250 - 10x

2. 900 - (10 × 25 + 10x)

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Using multiplication we know that there are 2,00,00,00,00,00,000 red blood cells in the body of a 125-pound person.

One of the four fundamental arithmetic operations, along with addition, subtraction, and division, is multiplication.

A product is the output of a multiplication operation.

When you take a single number and multiply it by several, you are multiplying.

We multiplied the number five by four times.

Due to this, multiplication is occasionally referred to as "times."

So, we have:

= 1.6 * 10¹¹

Solve as follows:

= 1.6 * 10¹¹

= (1.6 * 10¹¹) * 125

= (1.6 * 100000000000) * 125

= 1,60,00,00,00,000 * 125

= 2,00,00,00,00,00,000

Therefore, using multiplication we know that there are 2,00,00,00,00,00,000 red blood cells in the body of a 125-pound person.

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Answer:

a = 3
b = 8
c = 4

Step-by-step explanation:

We can identify the coefficients a, b, and c by comparing the equation with the standard form of a quadratic equation:

ax^2 + bx + c = 0

(always use this for these kinds of problems)

So, in the given equation:

The coefficient of x^2 is a = 3

The coefficient of x is b = 8

The constant term is c = 4

Answer:

(C) (x)^2 + (y)^2 = 49.

Step-by-step explanation:

The equation of a circle with center (h, k) and radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

In this case, the center of the circle is (0, 0), so h = 0 and k = 0. The radius of the circle is not given, but we can see that the equation must have a radius of 7 because the only terms involving x and y are squared and have coefficients of 1, which means they represent the distance from the center squared.

Therefore, the equation of the circle is:

(x - 0)^2 + (y - 0)^2 = 7^2

Simplifying, we get:

x^2 + y^2 = 49

So the correct answer is (C) (x)^2 + (y)^2 = 49.

The explaination of the volume of the cylinder calculation is below

To find the volume of a cylinder with a base radius of r units and a height of h units, you can use the formula:

V = πr²h

Where V is the volume of the cylinder, π is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder's base, and h is the height of the cylinder.

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The upper limit of the confidence interval for the proportion of millennials that care about immigration is 0.20.

In order to find the upper limit of the confidence interval for the proportion of millennials that care about immigration,

we first need to calculate the confidence interval using the margin of error (MOE) and sample proportion.

The formula to find the confidence interval is given by:

Confidence Interval = Sample Proportion ± Margin of Error (MOE)

Where Sample Proportion = Poll Result / 100

Let's first find the Sample Proportion of millennials that care about immigration.

Sample Proportion = Poll Result / 100= 17 / 100= 0.17

Now let's calculate the confidence interval using the MOE.

Confidence Interval = Sample Proportion ± Margin of Error (MOE)

Upper Limit of Confidence Interval = Sample Proportion + MOE= 0.17 + 0.03= 0.20

Therefore, the upper limit of the confidence interval for the proportion of millennials that care about immigration is

0.20.

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Answer:

-12.762538417

Step-by-step explanation:

The possible whole-number values of 'a' that satisfy the triangle inequality are 2, 3, 4, 5, and 6.

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

To create a triangle with sides a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side. This can be represented by the triangle inequality:

a + b > c

a + 6 > 7 (substituting the given values for b and c)

a > 1

b + c > a

6 + 7 > a

13 > a

a + c > b

a + 7 > 6

a > -1

Since the length of a cannot be negative, the inequality can be simplified to:

a > 1

Therefore, to create a triangle with side lengths of 7, 6, and some whole number value for a, a must be greater than 1. The possible whole-number values of 'a' that satisfy the triangle inequality are 2, 3, 4, 5, and 6.

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