What is a solid that has one vertex one flat face and can roll

Answer:

[tex]\large\boxed{x=4\pm2\sqrt6}[/tex]

Step-by-step explanation:

[tex]x^2-6x+12=2x+20\qquad\text{subtract}\ 2x\ \text{and}\ 12\ \text{from both sides}\\\\x^2-8x=8\\\\x^2-2(x)(4)=8\qquad\text{add}\ 4^2=16\ \text{to both sides}\\\\x^2-2(x)(4)+4^2=8+4^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-4)^2=24\iff x-4=\pm\sqrt{24}\\\\x-4=\pm\sqrt{4\cdot6}\\\\x-4=\pm\sqrt4\cdot\sqrt6\\\\x-4=\pm2\sqrt6\qquad\text{add 4 to both sides}\\\\x=4\pm2\sqrt6[/tex]

Final answer:

The quadratic equation x² - 6x + 12 = 2x + 20 is solved by completing the square, resulting in the solution x = 4 ± √24.

Explanation:

To solve the quadratic equation x² - 6x + 12 = 2x + 20 by completing the square, we first need to rearrange the equation to get 0 on one side. We do this by subtracting 2x and 20 from both sides, which gives us:

x² - 8x - 8 = 0

Next, we find the value to complete the square for the x terms. We take half of the coefficient of x, which is 8/2 = 4, and then square it, getting 4² = 16. We add 16 to both sides of the equation to form a perfect square on the left:

x² - 8x + 16 = 8 + 16

This simplifies to:

(x - 4)² = 24

To find the value of x, we take the square root of both sides:

x - 4 = ±√24

Finally, we solve for x:

x = 4 ± √24

Final answer:

To find the number of games won and lost by the baseball team, use the given win to loss ratio of 4/3 and the total number of games played (147) to establish a system of equations and solve for the number of wins and losses. The team won 84 games and lost 63 games.

Explanation:

The student has asked how many games a baseball team won and lost given that they played 147 regular season games with a win to loss ratio of 4/3. To solve this, let w represent the number of games won and l represent the number of games lost. According to the ratio, we have w/l = 4/3. Because the total number of games is 147, we can express this as w + l = 147.

First, express the number of losses in terms of wins using the ratio: l = 3w/4.

Next, substitute l in the total games equation: w + 3w/4 = 147.

Solving for w gives w = 84 (number of games won).

To find l, substitute w back into l = 3w/4 to get l = 63 (number of games lost).

The baseball team won 84 games and lost 63 games.

The minimum volume of hot air needed to lift a hot air balloon carrying 800 kg is 3200 m³.

The minimum volume of hot air needed to lift a hot air balloon carrying 800 kg can be calculated by using the density difference between hot air and cool air. Given that the density of hot air is about 0.25 kg/m³ less than that of cool air, we can calculate the volume of hot air needed.

Let V be the volume of hot air needed. The weight of the hot air balloon is equal to the weight of the cool air displaced by the hot air balloon. So we can set up the equation 800 kg = ((V + 1) - V) x 0.25 kg/m³, where 1 represents the volume of the hot air balloon itself.

Simplifying the equation, we have 800 kg = 0.25 kg/m x 1 m³, which gives us V = 800/0.25 = 3200 m³. Therefore, the minimum volume of hot air needed to lift the hot air balloon carrying 800 kg is 3200 m³.

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To find the width of one campsite, we subtract the area of the beach from the total area and divide by 5 (the number of campsites). We set up a quadratic equation and solve for the width, finding that it is approximately 13.93 yd.

The student's question is about finding the width of one campsite in a linear arrangement of five campsites along a beach. Given that the area of the campground, including the beach, is 950 sq yd and the distance from the edge of a campsite to the water is 4 yd, we can set up an equation to find the total width of the five campsites. The total width of the campsites will be the total area minus the area of the beach, divided by the length of the campsites (which is 5 times the width of one campsite).

Let's call the width of one campsite 'w'. The area of the beach is 5w * 4 yd, since it extends the entire length of the campsites and is 4 yd wide. Therefore, the area of just the campsites is 950 sq yd - 5w * 4 yd. Since we have five campsites, all of equal width, the total width of the campsites is 5w. So we can set up the following equation:

950 - 5w * 4 = 5w2

Now solve for w:

B. 13.93 yd. The approximate width of one campsite is 13.81 yards.

To find the width of one campsite, we need to divide the area of the campground by the number of campsites. The given area of the campground is 950 sq yd, and there are 5 campsites. So, the area of one campsite is 950 sq yd / 5 = 190 sq yd. Since the campsite is square, the width and length are equal. Let's assume the width of one campsite is 'x' yards. Then the equation becomes x^2 = 190. By taking the square root of both sides, we get x = √190 ≈ 13.81 yards. Therefore, the approximate width of one campsite is 13.81 yards.

The fraction of a batch of trail mix consisting of peanuts is 43/6.

A fraction is written in the form of a numerator and a denominator where the denominator is greater that the numerator.

Example: 1/2, 1/3 is a fraction.

We have,

Number of cups for peanuts = 2(2/3)

Total number of cups in the trial mix.

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

= 8/3 + 3/4 + 5/2 + 5/4

= (32 + 9 + 30 + 15)/12

= 86/12

= 43/6

= 7(1/6)

Thus,

The fraction of peanuts is 43/6.

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Final answer:

Each loaf of bread from the Westside bakery contains 0.44 pounds of flour.

Explanation:

To determine the amount of flour used in each loaf of bread, we need to divide the total amount of flour used by the number of loaves produced. In this case, the bakery uses 440 pounds of flour to make 1,000 loaves of bread. So, to find out the amount of flour used in each loaf, we divide 440 by 1,000.

440 ÷ 1,000 = 0.44 pounds of flour

Therefore, each loaf of bread contains 0.44 pounds of flour.

Final answer:

If an integer n, when divided by 7, leaves a remainder of 3, then multiplying n by 3 and dividing by 7 will leave a remainder of 2.

Explanation:

Let's solve the problem where n when divided by 7 leaves a remainder of 3 and find the remainder when 3n is divided by 7. According to the information, if n divided by 7 leaves a remainder of 3, we can express n as 7k + 3 for some integer k.

To find the remainder when 3n is divided by 7, substitute 7k + 3 for n: 3n = 3(7k + 3) = 21k + 9. We can further simplify this to 21k + 7 + 2, or 7(3k + 1) + 2. Here, it's evident that when 21k + 9 (or similarly 7(3k+1)+2) is divided by 7, it leaves a remainder of 2.

Answer:

[tex]V=\frac{27}{64}yd^3[/tex]

Step-by-step explanation:

The edge length of the cube shaped bin is [tex]\frac{3}{4}\text{ yards}[/tex]

Hence, [tex]s=\frac{3}{4}\text{ yards}[/tex]

The volume of the cube is given by

[tex]V=s^3[/tex]

Plugging the value of s, we get

[tex]V=(\frac{3}{4})^3[/tex]

This can be further simplified as

[tex]V=\frac{3\times3\times3}{4\times4\times4}\\\\V=\frac{27}{64}[/tex]

Hence, the volume of the container is 27/64 cubic yards.

Answer:

Step-by-step explanation:

12

You are given a table

[tex]\begin{array}{cc}\text{Number of books sold} & \text{Profit}\\100 & \$50\\250 & \$275\\300 & \$350\\350 & \$450\end{array}[/tex]

The rate of change for the function represented in the table can be calculated using the formula

[tex]\dfrac{y_{i+1}-y_i}{x_{i+1}-x_i},[/tex]

where i=1,2,3.

1. When i=1,

[tex]\dfrac{y_2-y_2}{x_2-x_1}=\dfrac{275-50}{250-100}=\dfrac{225}{150}=1.5[/tex]

2. When i=2,

[tex]\dfrac{y_3-y_2}{x_3-x_2}=\dfrac{350-275}{300-250}=\dfrac{75}{50}=1.5[/tex]

3. When i=3,

[tex]\dfrac{y_4-y_3}{x_4-x_3}=\dfrac{425-350}{350-300}=\dfrac{75}{50}=1.5[/tex]

Then the rate of change is $1.5

Answer: correct choice is D.

The rate of change for the function is [tex]\$ 1.5[/tex]

The rate  of change for the function is calculate by the formula

[tex]\frac{y_{n+1}-y_n}{x_{n+1}-x_n}[/tex]

Where [tex]n=1,2,3[/tex].

A) when [tex]n=1[/tex]

[tex]\frac{y_{1+1}-y_1}{x_{1+1}-x_1}=\frac{y_{2}-y_2}{x_{2}-x_2}[/tex]

[tex]\dfrac{275-50}{250-100}=\dfrac{225}{100}[/tex]

Hence Rate of change when [tex]n=1[/tex] is [tex]1.5[/tex]

B)when [tex]n=2[/tex]

[tex]\frac{y_{2+1}-y_2}{x_{2+1}-x_2}=\frac{y_{3}-y_2}{x_{3}-x_2}[/tex]

[tex]\dfrac{350-275}{300-250}=\dfrac{75}{50}[/tex]

Hence Rate of change when [tex]n=2[/tex] is [tex]1.5[/tex]

C) when [tex]n=3[/tex]

[tex]\frac{y_{3+1}-y_3}{x_{3+1}-x_3}=\frac{y_{4}-y_3}{x_{4}-x_3}[/tex]

[tex]\dfrac{425-350}{350-300}=\dfrac{75}{50}[/tex]

Hence Rate of change when [tex]n=3[/tex] is [tex]1.5[/tex]

So the rate of change for function represented in the table is same for different value of [tex]n[/tex] i.e, [tex]\$ 1.5[/tex].

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Answer:

5 boxes

Step-by-step explanation:

Given that;

The total amount Ray has with him to buy masonary nails and finishing nails for his project is $36

If Masonry nails sell for $5.50 per box

Finishing nails cost $3.50 per box.

Now, Only full boxes are available for purchase.

If Ray buys 3 boxes of masonry nails

i.e 3 × $5.50 = $16.5

what is the maximum number of boxes of finishing boxes he can buy?

To determine the number of finishing nails boxes he can buy. we will subtract the expense spent on the prior 3 masonary nails from his total amount

i.e $36 - $16.5

= $19.5

Let's not forget that, only full boxes are available for purchase and not half or quarter.

SO since 1 finishing nail box cost $3.50

then let x finishing nail box cost $19.5

∴ (x) × $3.50 =1 × $19.5

(x) = [tex]\frac{1*19.5}{3.50}[/tex]

(x)  = 5.57 boxes

Since full boxes are available for purchase then   (x)  ≅ 5 boxes, then Ray will be left with $2 at his pocket.

Therefore, the maximum number of boxes of finishing boxes he can buy is 5 boxes

Adding any rational number to 1/2 results in another rational number, because rational numbers can always be expressed as fractions with integers as numerators and denominators, making addition straightforward by finding a common denominator.

Any number that can be expressed as a fraction where both the numerator and denominator are integers (excluding zero as the denominator) will produce a rational number when added to 1/2. For example, adding 1/2 to 3 (which is the same as 3/1) will yield 3 1/2 or 7/2, still a rational number. By definition, rational numbers include integers, finite decimals, and repeating or terminating decimals, as they can all be represented as fractions.

Understanding Rational Numbers:

To solve the more difficult problem, one might need to find common denominators when working with more complex fractions. For instance, adding 1/2 to 2/3 requires a common denominator: 3/6 + 4/6 = 7/6. The key point here is that when you add any rational number to another rational number (like 1/2), the result will be rational because you can find a common denominator and then add the numerators, keeping that same common denominator.

Answer: the answer is c

Step-by-step explanation:

Step-by-step explanation:

Let's use this equation as an example - [tex]102=-7r+4[/tex]

The whole point of a linear equation is to find what the variable (which in our case is r ) equals to.

Our equation: [tex]102=-7r+4[/tex]

First step is to try to get r alone on one side.

First step: [tex]102-4=-7r[/tex] we wouldn't do [tex]102+4=-7r[/tex] (notice it's negative and not positive) since we're switching the positive 4 to the other side we'd switch the positive to a negative, making it [tex]102-4[/tex].

Know that everytime we switch a number or variable to the other side we must always switch the sign to the opposite of what it is. If it's dividing, we multiply, if we subtract, we add, etc. Same with the other way around.

Second step is to solve both sides of the equation.

Second step: [tex]98=-7r[/tex]

Third step is to divide the both sides by the number with the variable, making it 98 divided by -7r and -7r divided by -7

Third Step: [tex]\frac{98}{-7} =\frac{-7r}{-7}[/tex] = [tex]-14=r[/tex]

Now we've found our variable.

Answer: [tex]-14=r[/tex]

Answer:

i) The general formula to find the surface area of a prism = ph + 2B

ii) The area of the triangular prism = ph + 2([tex]\frac{1}{2} bh[/tex])

Step-by-step explanation:

A prism is a three-dimensional figure. The surface area of a figure is the sum of all the areas of its sides.

For example, rectangular prism has 6 sides, to find the surface of rectangular prism, we need to find the area of each side and add them together to get the surface area.

So, the general formula to find the surface area of a prism = ph + 2B, where "p" is the perimeter of the base, "h" is the height of the prism and B is the area of the base.

Now let's find the area of triangular prism.

In the triangular prism, there are two triangles and three rectangles.

Area of a triangle = [tex]\frac{1}{2} bh[/tex]

So, the area of the triangular prism = ph + 2([tex]\frac{1}{2} bh[/tex])

The value of the distance between the points (2, 3) and (2, 8) is,

⇒ d = 5 units

A pair of numbers which describe the exact position of a point on a cartesian plane by using the horizontal and vertical lines is called the coordinates.

Given that;

To find the distance between the points (2, 3) and (2, 8).

Since, The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Thus, The value of the distance between the points (2, 3) and (2, 8) is,

⇒ d = √(2 - 2)² + (8 - 3)²

⇒ d = √5²

⇒ d = 5 units

Thus, The value of the distance between the points (2, 3) and (2, 8) is,

⇒ d = 5 units

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