6. General questions 1. volume of pyramid b) progress , ideas • some people remembered a formula for volume of pyramid V = 1/3 lwh MAIN PROBLEM: where dues this come from? Why 1/3 ? • one strategy: dissect cube (orrectangule box) (assume for now l=w=h= 1) what is volume of negative space, compared to pyramid ? • second strategy glut multiple priemies together • third strategy: Slice pyramid horizontally Problems: • Keep thinking about strategy one and the • For strategy 3, fiad fald volume for four blocks that I drew above. (That is an overestimat) • Also we smiler stratey to set can underestimate That give range? • Try ti make more

The coordinates of N can be estimated as: (-8, -12)

Scale factor is a mathematical term, which we use to scale 2-dimensional and 3-dimensional shapes. Scale factor is a measure of similar figures. Those figures who look the same but are in different sizes.

If point N is dilated with a scale factor of 5/4 centered at the origin, the coordinates of the image point, let's call it N', can be found by multiplying the coordinates of N by the scale factor:

N' = (5/4)N

We also know that the image point N' is located at (-10, -15), so we can substitute these values into the equation:

(-10, -15) = (5/4)N

Solving for N, we can multiply both sides by the reciprocal of 5/4, which is 4/5:

N = (4/5)(-10, -15)

N = (-8, -12)

Therefore, the coordinates of point N are (-8, -12).

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Answer:

80

Step-by-step explanation:

we will write an inequality of 1.50x + 130 =250

We subtract 130 from each side, so we have

1.50x=120

Now we divide by 1.50 on each side

X=80.

For a dish party, Alia needs to seat four people at a round table. There are 24 possible combinations of seating arrangements.

This is because each person has 3 options (clockwise, counter-clockwise, opposite), for a total of 34 or 81 combinations. However, due to symmetry, half of these are the same combinations with the people just shifted around the table, so we are left with 24 total combinations.

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Answer: $135

Step-by-step explanation:

    We know that:

         Selling price = [tex](100 - \text{Loss}\%)(\frac{\text{Cost Price}}{100})[/tex]

    Using the given cost price and loss%, we can solve.

         Selling Price = [tex](100 - \text{Loss}\%)(\frac{\text{Cost Price}}{100})[/tex]

         Selling Price = (100 - 10)([tex]\frac{\text{150}}{100}[/tex])

         Selling Price = (90) * (1.5)

         Selling Price = $135

    This means the selling price of the dress is $135.

Some of the questions to ask grade 12 students on a questionnaire about taking a gap year after school​ are:

A questionnaire refers to a set of questions or items used to collect information about respondents' attitudes, experiences, or opinions. Questionnaires can be used to collect both quantitative and qualitative data. Questionnaires are widely used in market research, as well as social and health sciences.

As the researcher's study focus on semester taken after high school and prior to post-secondary education, such questionnaire involve will focus on variables on the student gap year.

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The equation of the function g(x) in terms of the function f(x) has the equation of g(x) = -2f(x + 2) + 3

Given the graph of functions f(x) and g(x)

Where

y = f(x) -- the parent function

y = g(x) --- the transformed function (the dashed line)

The transformation of f(x) to g(x) is as follows

First, the graph of y = f(x) is moved 2 units to the left

This is represented as f(x + 2)

Next, the shifted graph of y = f(x) is reflected across the x-axis and then stretched vertically by a factor of 2.

This is represented as -2f(x + 2)

Lastly, the reflected and stretched graph is moved 3 units up

This is represented as -2f(x + 2) + 3

This means that

g(x) = -2f(x + 2) + 3

So, the function g(x) is g(x) = -2f(x + 2) + 3

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So the equation of the circle is (x+4)^2 + (y+4)^2 = 4^2 with center (-4, -4) and radius 4

The equation of a circle is given by (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center of the circle and r is the radius. We need to rearrange the given equation to match this form.

2x^2 + 2y^2 + 16x + 16y + 32 = 0

Divide both sides of the equation by 2:

x^2 + y^2 + 8x + 8y + 16 = 0

Rearrange the terms:

x^2 + 8x + y^2 + 8y = -16

Complete the square for both x and y terms:

(x^2 + 8x + 16) + (y^2 + 8y + 16) = -16 + 16 + 16

(x+4)^2 + (y+4)^2 = 16

Now the equation is in the standard form of a circle equation, and we can find the center (h,k) and radius r:

h = -4, k = -4, r = 4

So the equation of the circle is (x+4)^2 + (y+4)^2 = 4^2 with center (-4, -4) and radius 4.

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We have shown that the log of the geometric mean of two numbers is the arithmetic mean of the logs of those two numbers.

We can use the properties of logarithms to show that the log of the geometric mean of two numbers is the arithmetic mean of the logs of those two numbers.

First, recall that the arithmetic mean of two numbers is half their sum:
arithmetic mean = (v + w)/2

And the geometric mean is the square root of their product:
geometric mean = √(v*w)

Now, let's take the log of the geometric mean:
log(geometric mean) = log(√(v*w))

Using the property of logarithms that log(a*b) = log(a) + log(b), we can expand the expression:
log(√(v*w)) = log(√v) + log(√w)

Using the property of logarithms that log(a^b) = b*log(a), we can simplify the expression:
log(√v) + log(√w) = (1/2)*log(v) + (1/2)*log(w)

Now, we can see that the log of the geometric mean is the arithmetic mean of the logs of the two numbers:
log(geometric mean) = (log(v) + log(w))/2

So, if we let p = log(v) and q = log(w), we can see that the log of the geometric mean of v and w is the arithmetic mean of p and q:
log(geometric mean) = (p + q)/2

Therefore, we have shown that the log of the geometric mean of two numbers is the arithmetic mean of the logs of those two numbers.

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The area of the shaded shape WXZY s equal to 381.4 cm².

Mathematically, the area of a sector can be calculated by using this formula:

Area of sector = θπr²/360

Where:

In order to determine the area of the shaded shape WXZY, we would have to subtract the area of sector OYZ from the area of the sector OWX.

Area of sector OWX = 75 × 3.14 × 32²/360 = 669.87 cm²

Area of sector OYZ = 75 × 3.14 × 21²/360 = 288.49 cm²

Area of shaded shape WXZY = Area of sector OWX - Area of sector OYZ

Area of shaded shape WXZY = 669.87 cm² - 288.49 cm²

Area of shaded shape WXZY = 381.4 cm².

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The solution is, (x + 6) and (2x - 5) are the factors of polynomial

2x^2 + 7x - 30.

Polynomials are sums of k-xⁿ terms, where k can be any number and n can be any positive integer.

3x+2x-5, for example, is a polynomial.

A binomial is a two-term polynomial.

For instance, x²ˣ-2ˣ² and x⁶ˣ-6ˣ⁶ are both binomials.

here, we have,

given that,

2x^2 + 7x - 30 is the polynomial .

now, we have to factorize it.

so, we get,

2x2 + 7x - 30

2x2 + 12x - 5x - 30

2x( x + 6) - 5( x + 6)

so, we have,

(x + 6) and (2x - 5) are the factors.

Hence, The solution is, (x + 6) and (2x - 5) are the factors of polynomial

2x^2 + 7x - 30.

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The values of x and y are determined by solving the equation

The values of x and y can be found by solving the equation x(23) + y(-10) = (106). This can be done by isolating one variable and solving for the other. For example, isolating x gives us x = (106 + 10y)/23. We can then plug this value of x back into the original equation and solve for y. Once we have the value of y, we can plug it back into the equation for x to find the value of x.

The value of −2p2s2v​−qtw​−rux​​ can be found by plugging in the given values of p, s, v, q, t, w, r, and u into the equation and simplifying.

The value of ∣3AB∣ can be found by using the properties of determinants of matrices. Specifically, the determinant of a product of matrices is equal to the product of the determinants of the individual matrices. Therefore, ∣3AB∣ = ∣3∣∣A∣∣B∣ = 3(-1)(3) = -9.

So, the values of x and y are determined by solving the equation x(23) + y(-10) = (106), the value of −2p2s2v​−qtw​−rux​​ is found by plugging in the given values and simplifying, and the value of ∣3AB∣ is found by using the properties of determinants of matrices.

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She gets $114 to work at grocery store for 6 hours and $200 at the nursery.

A rate in arithmetic is a ratio that contrasts two separate values with various unit systems. For instance, if John types 50 words per minute, that means he types 50 words per minute.

We are dealing with a rate because the word "per" is there. The symbol "/" can be used in place of the word "per" in issues.

When two or more similar amounts or numbers are being compared using the same units, a ratio is utilized. When referring to the ratio of one quantity "to" the second quantity in spoken language, it is frequently written with a colon.

Eli has two summer jobs. During the week he works in the grocery store, and on the weekend he works at a nursery. He gets paid $20 per hour to work at the grocery store and $17 per hour to work at the nursery.

14 hours at the grocery store and 9 hours at the nursery? How much does he earn if he works

Therefore, She gets $114 to work at grocery store for 6 hours and $200 at the nursery.

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Answer: y > 2/5x - 4

Step-by-step explanation:

The slope intercept form is y = mx + b

To get the inequality to that, we have to get y alone.

Starting:

2x - 5y < 20

Subtract 2x from each side leaves you:

- 5y < -2x + 20

Dividing by -5 leaves you:

(Remember that when dividing by a negative in an inequality, the sign flips.)

y > 2/5x - 4

Hope this helps!

Function Family: quadratic

Domain: all real numbers

x-int: (0,0) and (2,0)

y-int: (0,0)

The GCF (greatest common factor) of the first two terms of the polynomial 3p^(3)+9p^(2)+5p+15 is 3p^(2). This is because both terms have a common factor of 3p^(2) that can be factored out.

The GCF of the last two terms of the polynomial is 5. This is because both terms have a common factor of 5 that can be factored out.

Therefore, the polynomial can be factored as follows:

3p^(2)(p+3) + 5(p+3)

Now, we can factor out the common factor of (p+3) to get the final factored form of the polynomial:

(3p^(2)+5)(p+3)

So, the GCF of the first two terms is 3p^(2) and the GCF of the last two terms is 5.

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The object will have a height of 100 feet after 100 seconds.

The equation for the height of the object in feet after T seconds is H(T) = A(T - h)2 + k, where A and h are constants. Given that the object has a height of 0 feet after 10 seconds, and a maximum height of 200 feet at 0 seconds, we can solve for A and h.

We know that H(0) = 200, so k = 200.

We also know that H(10) = 0, so A(10 - h)2 + 200 = 0. Solving for A, we get A = -(h2 + 200) / 102.

Now we have an equation with two unknowns, A and h, so we can use substitution to solve for h. We can substitute -(h2 + 200) / 102 for A in our original equation, and solve for h. This gives us h = 10.

We now know A = -(102 + 200) / 102 = -20.

So, our equation for the height of the object in feet after T seconds is H(T) = -20(T - 10)2 + 200.

To find the height of the object after 100 seconds, substitute 100 for T in the equation. This gives us H(100) = -20(100 - 10)2 + 200 = 100.

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The correct statement about the functions f(x) and g(x) include the following: B. f(x) has a steeper slope than g(x).

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Substituting the given points into the slope formula, we have the following;

Slope, m of g(x) = (1 - (-3))/(0 - (-2))

Slope, m of g(x) = 4/2

Slope, m of g(x) = 2.

Slope of f(x) = 3. Therefore, 3 is greater than 2.

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Answer:

Step-by-step explanation:

the answer is eitehr b or a

Mason spends one half of an hour lifting weights and one third of an hour on the treadmill, so the ratio of time Mason spends lifting weights to time Mason spends on the treadmill is 1:2.

A treadmill is a machine used for walking or running while staying in one place. It usually has a belt and a platform to stand on, and the belt moves at a speed determined by the user. It is a great way to get a good cardio workout.

The question asks which are correct descriptions of the relationship of time Mason spends lifting weights and on the treadmill.

Mason spends one half of an hour lifting weights and one third of an hour on the treadmill, so the fraction of time Mason spends lifting weights is 2/3.

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The price per pound of the apples is $3.50. This is calculated by dividing the total cost, $7.00, by the weight of the apples, 2 pounds.

The price per pound of the apples can be found by dividing the total cost by the weight of the apples. In this example, the total cost of the apples is $7.00 and the weight is 2 pounds. To calculate the price per pound, divide 7.00 by 2. This gives us 3.50, which is the price per pound of the apples. Knowing the price per pound is useful when you are trying to figure out how much a certain amount of apples will cost. For example, if you wanted to buy 5 pounds of apples, you would multiply 5 by 3.50, which would give you $17.50. This is the total cost of the 5 pounds of apples. In addition to helping you figure out how much certain amounts of apples will cost, knowing the price per pound can also help you compare prices between stores. For example, if you found apples for $4.00 per pound at one store, and $3.50 per pound at another store, you could easily determine which store has the better deal.

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The expression (122two _ 202three)_ 1001three can be evaluated by breaking it down into its constituent parts.

Expression in math is a combination of numbers, symbols, and/or operations that represent a mathematical statement. Expressions can be used to calculate values, solve equations, and to model real-world problems. Expressions can be simple or complex and can range from basic arithmetic operations to more complex functions and equations. Expressions are an essential part of mathematics and are used to represent equations, formulas, and even solutions to complex problems.

The first part, 122two, is a number and can be evaluated to 122. The second part, 202three, is also a number and can be evaluated to 203. The third part, 1001three, is also a number and can be evaluated to 1002. Taking the evaluation of each part (122, 203 and 1002), the overall expression (122two _ 202three)_ 1001three can be evaluated to 121.

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(a) (6, H, 7) is not a possible outcome because the sum of the two numbers rolled on the number cube cannot be 7 if the first number rolled is 6.

(b) (T, 3, 4) is a possible outcome because the first outcome is tails, the second outcome is 3, and the third outcome is 4.

(c) (6, H, 6, T) is not a possible outcome because the third outcome cannot be both 6 and T at the same time.

(d) (1, T, 1) is a possible outcome because the first and third outcomes are both 1 and the second outcome is tails.

(e) (H, H, 3) is not a possible outcome because the third outcome cannot be 3 if the first two outcomes are both heads.

What is Probability?

Probability is a branch of mathematics in which the chances of experiments occurring are calculated. It is by means of a probability, for example, that we can know from the chance of getting heads or tails in the launch of a coin to the chance of error in research.

Since each of the three people orders a different flavor, the first person has 6 choices, the second person has 5 choices (since one flavor has already been chosen), and the third person has 4 choices (since two flavors have already been chosen). The total number of outcomes is the product of these numbers:

6 × 5 × 4 = 120

Therefore, there are 120 possible outcomes.

(part 1) There are 10 options for the first digit, 9 options for the second digit (since one digit has already been chosen), and 8 options for the third digit (since two digits have already been chosen). The total number of combinations is the product of these numbers:

10 × 9 × 8 = 720

Therefore, there are 720 possible locker combinations.

(part 2) If you absolutely choose 9 as your first digit, then there are only 9 options for the first digit. The second and third digits can still be any of the remaining 9 digits (since digits cannot repeat). Therefore, the number of possible combinations is:

1 × 9 × 8 = 72

The theoretical probability of getting HH on two coin flips is (1/2) × (1/2) = 1/4 = 0.25. The experimental probability of getting HH after 100 trials is 25/100 = 0.25. Since the experimental probability is equal to the theoretical probability, we can say that the experimental probability of getting HH is equal to the theoretical probability.

The possible outcomes of rolling a standard number cube are 1, 2, 3, 4, 5, and 6. The possible outcomes of flipping a coin are heads (H) or tails (T).

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The chain rule allows us to reduce the first equation to the second equation by finding the derivative of the composite function and substituting it for the original function. This provides a simpler and more concise explanation of the relationship between the two equations.

The chain rule is a formula for calculating the derivative of a composite function. It states that the derivative of a composite function f(g(x)) is equal to the derivative of f with respect to g times the derivative of g with respect to x. In mathematical notation, this is expressed as (f(g(x)))' = f'(g(x)) * g'(x).

In the case of the first equation, we can use the chain rule to reduce it to the second equation by applying the formula mentioned above. First, we need to identify the composite function and its components. In this case, the composite function is f(g(x)) and the components are f and g.

Next, we need to find the derivative of f with respect to g and the derivative of g with respect to x. Once we have these derivatives, we can multiply them together to get the derivative of the composite function.

Finally, we can reduce the first equation to the second equation by substituting the derivative of the composite function for the original composite function. This will give us the second equation, which is a simplified version of the first equation.

In conclusion, the chain rule allows us to reduce the first equation to the second equation by finding the derivative of the composite function and substituting it for the original function. This provides a simpler and more concise explanation of the relationship between the two equations.

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The exact value of the expression is √3.

To find the exact value of the given expression, we can use the sum formula for tangent. The formula is:

Tan (A + B) = (tan A + tan B) / (1 - tan A tan B)

Where A and B angle.

In this case, A = 41° and B = 19°. Plugging these values into the formula, we get:

Tan (41° + 19°) = (tan 41° + tan 19°) / (1 - tan 41° tan 19°)Simplifying the left side of the equation gives us:

Tan 60° = (tan 41° + tan 19°) / (1 - tan 41° tan 19°)We know that tan 60° = √3.

Substituting this value into the equation gives us:√3 = (tan 41° + tan 19°) / (1 - tan 41° tan 19°)

Cross-multiplying and simplifying gives us:

√3 (1 - tan 41° tan 19°) = tan 41° + tan 19°√3 - √3 tan 41° tan 19° = tan 41° + tan 19°

We can rearrange the terms to get:

√3 = tan 41° + tan 19° + √3 tan 41° tan 19°

This is the same expression that we were given in the question.

Therefore, the exact value of the expression is √3.

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Uncle Moneybags can withdrawal approximately $583.67 per month.

To find out how much Uncle Moneybags can withdraw each month, we can use the formula for the future value of an annuity:

FV = PMT × [(1 + i)ⁿ - 1] / i

Where:
FV = future value
PMT = payment amount
i = interest rate per period
n = number of periods

In this case, we want to solve for PMT, so we can rearrange the formula to:

PMT = FV × i / [(1 + i)ⁿ - 1]

We know that FV = $500,000, i = 8% / 12 = 0.00667, and n = 25 × 12 = 300. Plugging these values into the formula, we get:

PMT = $500,000 × 0.00667 / [(1 + 0.00667)³⁰⁰ - 1]

PMT = $500,000 × 0.00667 / 5.7435

PMT = $583.67

Therefore, Uncle Moneybags can withdraw $583.67 each month for 25 years.

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The values of x and y that satisfy the given equation are x = 2 and y = 1.

One or many equations having the same number of unknowns that can be solved simultaneously are called simultaneous equations. And the simultaneous equation is the system of equations.

Since a and b span R², any vector in R² can be expressed as a linear combination of a and b.

Let's call the coefficients of this linear combination c₁ and c₂ so that any vector v in R² can be expressed as:

v = c₁a + c₂b

Now let's use this fact to solve the given equation:

(2-x)a + b = ya + (x-3)b

We can rearrange this equation to get all the terms with an on one side and all the terms with b on the other side:

(2-x)a - ya = (3-x)b

Now we can write both sides of the equation as linear combinations of a and b:

(2-x)a - ya = 2a - xa - ya = (2-x)a - (x-2)a - ya = (2-x - x + 2)a - ya = (4 - 2x)a - ya

(3-x)b = 3b - xb = 3b - xb + 3b - 3b = (6 - x)b - 3b = (6 - x - 3)b = (3 - x)b

So now we have:

(4 - 2x)a - ya = (3 - x)b

Since a and b span R², we know that any vector in R² can be expressed as a linear combination of a and b. Therefore, we can write:

(4 - 2x)a - ya = c₁a + c₂b

(3 - x)b = c₁a + c₂b

Now we can set the coefficients of a and b equal to each other and solve for x and y:

4 - 2x = c₁

-y = c₂

3 - x = c₂

Substituting the third equation into the second equation, we get:

-y = 3 - x

Solving for x and y, we get

x = 2

y = 1

Therefore, the values are x = 2 and y = 1.

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From  the function φ(x)=31​x3−21​x,  0 is a fixed point of φ(x) it shows in calculation below.  The iteration xk+1 = φ(xk) converges locally to 0 shows below. The iteration diverges as k increases.

(a) To show that 0 is a fixed point of φ(x), we need to show that φ(0) = 0. Substituting x = 0 into the function gives us:

φ(0) = (3/1)(0)3 - (2/1)(0) = 0 - 0 = 0

Therefore, 0 is a fixed point of φ(x).

(b) To show that the iteration xk+1 = φ(xk) converges locally to 0, we need to show that |φ'(x)| < 1 for x near 0. Taking the derivative of φ(x) gives us:

φ'(x) = (9/1)x2 - (2/1)

At x = 0, φ'(0) = 0 - 2 = -2. Since |φ'(0)| > 1, the iteration does not converge locally to 0.

(c) By numerical tests, we can find an initial guess x0 such that the iteration converges to 0 and an initial guess x0 such that the iteration diverges. For example, if we choose x0 = 0.5, the iteration converges to 0:

x1 = φ(0.5) = (3/1)(0.5)3 - (2/1)(0.5) = -0.625

x2 = φ(-0.625) = (3/1)(-0.625)3 - (2/1)(-0.625) = 0.8203125

x3 = φ(0.8203125) = (3/1)(0.8203125)3 - (2/1)(0.8203125) = -0.4625244140625

And so on. The iteration converges to 0 as k increases.

On the other hand, if we choose x0 = 2, the iteration diverges:

x1 = φ(2) = (3/1)(2)3 - (2/1)(2) = 20

x2 = φ(20) = (3/1)(20)3 - (2/1)(20) = 23980

x3 = φ(23980) = (3/1)(23980)3 - (2/1)(23980) = 3.47119332792e+13

And so on. The iteration diverges as k increases.

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