Samantha is designing a new three-dimensional puzzle in the form of a square pyramid. She knows that the design and the general formula for finding the volume are as follows, where h is the height and a is the side length of the base. All of her measurements are in centimeters.

An illustration of a pyramid, with the height of h and width a.

Samantha's puzzle must have a certain volume and height, but she needs to determine the side length of the base to complete the design. Which equation can Samantha use to find the side length of the base in terms of the volume and height of her puzzle?

Answer:

Step-by-step explanation:

vertex form = y=a(x-h)^2+k

standard form = y=ax^2+bx+c

the most straightforward way to solve it is to expand the vertex form

we get :

-(x+1)(x+1)-1

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

note: remember to leave the negative sign out of the parentheses and distribute it after - otherwise you may mix up signs

= (-x^2-x-x-1) - 1

= -x^2-2x-1-1

= -x^2 - 2x - 2

So, in standard form, it is y=-x^2-2x-2

Hope this helps!

Answer:

7

Step-by-step explanation:

We can use the fact that the corresponding sides of similar triangles are in proportion to each other.

Let's compare the corresponding sides of the two triangles:

Corresponding sides and ratio

Ratio

Since the triangles are similar, we know that these ratios are equal. That is,

6 / (x-3) = 12 / (x+1)

We can simplify this equation by cross-multiplying:

6(x+1) = 12(x-3)

Expanding and simplifying:

6x + 6 = 12x - 36

42 = 6x

x = 7

Therefore, x has a value of 7.

Using the given random variable and the mean the required probability in the given situation is 0.9772.

A probability is a numerical representation of the likelihood or chance that a specific event will take place.

Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

= P(x>36) = P(X-μ/σ>36-50/7)

= P(Z>-14/7) Z=X-μ/σ

= P(Z>-2)

= P(Z<2)  [P(Z<z) = P(Z>-z)]

= 0.9772 by the p-value table

Therefore, using the given random variable and the mean the required probability in the given situation is 0.9772.

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Correct question:

Assume the random variable x is normally distributed with mean 50 and a standard deviation 7. Find the indicated probability. P(X>36)

Area of sector XYZ is 81.45 feet².

The area of a sector is a measure of the size of a portion of a circle enclosed by two radii and an arc between them. It is expressed in square units, such as square centimeters, square meters, or square inches.

To find the area of a sector, you need to know the radius of the circle and the central angle of the sector.

The formula for the area of a sector is:

Area of sector = (central angle / 360°) x π x r²

where r is the radius of the circle, π is the mathematical constant pi (approximately 3.14), and the central angle is measured in degrees.

n is the area of sector XYZ

n/360=115/255(X)

n/255=115/360(V)

(The same elements are proportional)

n=115/360×255

n=81.4583≈81.45 feet²

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Answer:

A) angle 1 is congruent (equal size) to angle 4.

Step-by-step explanation:

when 2 lines intersect, then the intersection angles are the same on both sides of any of the lines. they are only left-right mirrored.

Answer:

A) angle 1 is congruent (equal size) to angle 4.

Step-by-step explanation:

Answer:

[tex]f(2\pi) = 32\pi^3 - 2[/tex]

Step-by-step explanation:

Main steps:

Step 1: Use integration to find a general equation for f

Step 2: Find the value of the constant of integration

Step 3: Find the value of f for the given input

Step 1: Use integration to find a general equation for f

If [tex]f'(x) = g(x)[/tex], then [tex]f(x) = \int g(x) ~dx[/tex]

[tex]f(x) = \int [12x^2 - sin(x)] ~dx[/tex]

Integration of a difference is the difference of the integrals

[tex]f(x) = \int 12x^2 ~dx - \int sin(x) ~dx[/tex]

Scalar rule

[tex]f(x) = 12\int x^2 ~dx - \int sin(x) ~dx[/tex]

Apply the Power rule & integral relationship between sine and cosine:

[tex]f(x) = 12*(\frac{1}{3}x^3+C_1) - (-cos(x) + C_2)[/tex]

Simplifying

[tex]f(x) = 12*\frac{1}{3}x^3+12*C_1 +cos(x) + -C_2[/tex]

[tex]f(x) = 4x^3+cos(x) +(12C_1 -C_2)[/tex]

Ultimately, all of the constant of integration terms at the end can combine into one single unknown constant of integration:

[tex]f(x) = 4x^3 + cos(x) + C[/tex]

Step 2: Find the value of the constant of integration

Now, according to the problem, [tex]f(0) = -2[/tex], so we can substitute those x,y values into the equation and solve for the value of the constant of integration:

[tex]-2 = 4(0)^3 + cos(0) + C[/tex]

[tex]-2 = 0 + 1 + C[/tex]

[tex]-2 = 1 + C[/tex]

[tex]-3 = C[/tex]

Knowing the constant of integration, we now know the full equation for the function f:

[tex]f(x) = 4x^3 + cos(x) -3[/tex]

Step 3: Find the value of f for the given input

So, to find [tex]f(2\pi)[/tex], use 2 pi as the input, and simplify:

[tex]f(2\pi) = 4(2\pi)^3 + cos(2\pi) -3[/tex]

[tex]f(2\pi) = 4*8\pi^3 + 1 -3[/tex]

[tex]f(2\pi) = 32\pi^3 - 2[/tex]

Answer:

[tex]f(2 \pi)=32\pi^3-2[/tex]

Step-by-step explanation:

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given:

If g(x) is the first derivative of f(x), we can find f(x) by integrating g(x) and using f(0) = -2 to find the constant of integration.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]     [tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $\sin x$}\\\\$\displaystyle \int \sin x\:\text{d}x=-\cos x+\text{C}$\end{minipage}}[/tex]

[tex]\begin{aligned} \displaystyle f(x)&=\int f'(x)\; \text{d}x\\\\&=\int g(x)\;\text{d}x\\\\&=\int (12x^2-\sin x)\;\text{d}x\\\\&=\int 12x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\int x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\cdot \dfrac{x^{(2+1)}}{2+1}-(-\cos x)+\text{C}\\\\&=4x^{3}+\cos x+\text{C}\end{aligned}[/tex]

To find the constant of integration, substitute f(0) = -2 and solve for C:

[tex]\begin{aligned}f(0)=4(0)^3+\cos (0) + \text{C}&=-2\\0+1+\text{C}&=-2\\\text{C}&=-3\end{aligned}[/tex]

Therefore, the equation of function f(x) is:

[tex]\boxed{f(x)=4x^3+ \cos x - 3}[/tex]

To find the value of f(2π), substitute x = 2π into function f(x):

[tex]\begin{aligned}f(2 \pi)&=4(2 \pi)^3+ \cos (2 \pi) - 3\\&=4\cdot 2^3 \cdot \pi^3+1 - 3\\&=32\pi^3-2\\\end{aligned}[/tex]

Therefore, the value of f(2π) is 32π³ - 2.

Answer: The man will make deposits for 66 - 45 = 21 years, or 21 x 4 = 84 quarters.

The quarterly interest rate is 4% / 4 = 1%.

Let's use the formula for the future value of an annuity:

FV = PMT x ((1 + r)^n - 1) / r

where FV is the future value of the annuity, PMT is the periodic payment, r is the interest rate per period, and n is the number of periods.

In this case, PMT = $2500, r = 1%, and n = 84. Substituting these values into the formula, we get:

FV = $2500 x ((1 + 0.01)^84 - 1) / 0.01

FV = $2500 x (5.409 - 1) / 0.01

FV = $2500 x 540.9

FV = $1,352,250

Therefore, there will be $1,352,250 in the account when the man retires at age 71.

The value of x it the nearest tenth of degree is 25.6°

The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.

Sin(tetha) = opp/ hyp

cos( tetha) = adj/ hyp

tan( tetha) = opp/ adj

Here, the hypotenuse is 92 and adjascent to the to the angle x is 63

therefore to calculate x we use cosine function

cos x = 83/92

cos x = 0.902

x = cos^-1( 0.902)

x = 25.6° ( nearest tenth)

therefore the value of x is 25.6°

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The measure of length of x of the parallelogram is x = 2.36 units

Given data ,

Let the parallelogram be represented as ABCD

Now , the measure of lengths of the sides are

The measure of AB = 13x - 20

The measure of CD = 2x + 6

The measure of BC = 5x + 4

Opposite sides are parallel

Opposite sides are congruent

So , AB = CD

13x - 20 = 2x + 6

Subtracting 2x on both sides , we get

11x = 26

Divide by 11 on both sides , we get

x = 2.36 units

Hence , the parallelogram is solved and x = 2.36 units

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The lateral and total surface area of the prism are 1000 square feet and 1120 square feet.

In this problem we need to determine the lateral and total surface area of a prism with a triangular base. The lateral surface area is the sum of the areas of the three rectangles and the total surface area is the sum of the lateral surface area and the areas of the two triangles.

The area formulas for the triangle and the rectangle are, respectively:

Triangle:

A = s · √[s · (s - a) · (s - b) · (s - c)]

s = 0.5 · (a + b + c)

Where:

Rectangle:

A = w · h

Where:

Now we proceed to determine each surface area:

Lateral surface area:

A = 2 · (20 ft) · (17 ft) + (20 ft) · (16 ft)

A = 1000 ft²

Total surface area:

s = 0.5 · (17 ft + 17 ft + 16 ft)

s = 25 ft

A = √[(25 ft) · (25 ft - 17 ft)² · (25 ft - 16 ft)] + 1000 ft²

A = 1120 ft²

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Answer:

40 degrees and 140 degrees

Step-by-step explanation:

To solve this problem you can add x+3x+20 and set that equal to 180. (We can do this because angle x and angle 3x + 20 make a linear pair. Knowing this we can estimate that both angles added together will equal 180)

Let us add x + 3x + 20 = 180 to find x. We can then substitute that into the equation.

[tex]x+3x+20=180 :a\\\\4x+20=180 :b\\\\4x + 20 -20=180-20:c\\\\4x=160:d\\\\\frac{4x}{4} = \frac{160}{4}:e\\\\x=40:f[/tex]

a: So in this part, we have rewritten the equation to make it easier to solve

b:  In this step, you combine the like terms x+3x to get 4x

c: In step c, you are subtracting 20 from both sides to keep constants on one side and variables on the right

d: In this last step the equation has been simplified to make it easier to solve.

e: To isolate x you have to divide both sides by 4, we do this because the coefficient of x is 4 so you divide the equation by 4 to cancel it out.

f: You rewrite and simplify the equation.

Now to find the measure of both angles you substitute x into the equation.

The first angle's value is 40 degrees and the second is 140 degrees.

These are our answers.

The interest for the month is:  $2.59

The balance for the month is: $598.18.

We have the information from the question is:

The interest rate is 1.4% per month.

June 1 Balance $352.12

June 4 Sears $331.89

June 8 eBay $81.58

June 15 Outback $30

June 18 Payment $200

Now, According to the question:

The average daily balance for June is $184.89.

To calculate the interest for the month,

Multiply the average daily balance by the interest rate:

$184.89 × 1.4% = $2.59.

To find the balance for the month :

Add the initial balance, transactions, and interest, then subtract the payment:

$352.12 + $331.89 + $81.58 + $30 + $2.59 - $200 = $598.18.

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Answer:

2 cans

Step-by-step explanation:

If she can give out 12 to each of her friends and have 3 left over. or she can give 9 out to each of her friends and have 12 left over, Malaika has 3 friends who can receive candies.

Let's suppose Malaika has "c" candies, and "f" is the number of friends she has.

According to the problem statement, we have two equations:

c = 12f + 3

c = 9f + 12

To solve for "f", we can set the two equations equal to each other:

12f + 3 = 9f + 12

Simplifying the equation, we get:

3f = 9

Dividing both sides by 3, we get:

f = 3

We can verify our solution by plugging "f=3" back into one of the original equations:

c = 12f + 3

c = 12(3) + 3

c = 39

So, Malaika has 39 candies in total. We can also check the other equation to make sure it is true:

c = 9f + 12

c = 9(3) + 12

c = 39

Both equations are true, so our solution of "f=3" is correct.

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The following expression is equivalent to [tex]x^{-5/3}[/tex] as option B that is 1/∛x⁵.

A fraction is a numerical representation of a part or a portion of a whole. It is expressed as one integer (numerator) divided by another integer (denominator), separated by a horizontal line.

Here,

We can rewrite [tex]x^{-5/3}[/tex]as [tex](1/ x^{^(5/3)})[/tex].

The negative exponent in the numerator tells us that we need to move the term to the denominator of a fraction and change the sign of the exponent to make it positive. Similarly, the fractional exponent in the denominator indicates that we need to take the reciprocal of the term and change the sign of the exponent to make it positive.

Therefore, we can rewrite [tex]x^{-5/3}[/tex] as:

[tex](1/ x^{^(5/3)})[/tex]

or

[tex]x^{-5/3}[/tex] = [tex](1/ x^{^(5/3)})[/tex]

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Complete question:

Which of the following is equivalent to [tex]x^{-5/3}[/tex]?

Answer:

Step-by-step explanation:

-9+6-1-2-3-4+5-6

-3-3-7-1

-6-8

-14

Answer:

The answer is -14.

Step-by-step explanation:

-(9 - 6) + (-1 -2) - (3 + 4) + (5 - 6)

Subtract 6 from 9 to get 3.

-3 -1 -2 - (3 + 4) + 5 - 6

Subtract 1 from -3 to get -4.

-4 - 2 - (3 + 4) + 5 - 6

Subtract 2 from -4 to get -6.

-6 - (3 + 4) + 5 - 6

Add 3 and 4 to get 7.

-6 - 7 + 5 - 6

Subtract 7 from -6 to get -13.

-13 + 5 - 6

Add -13 and 5 to get -8.

-8 - 6

Subtract 6 from -8 to get -14.

-14.

The probability that event A or B takes place is P ( A ∪ B ) = 6/17

Given data ,

Let the probability that event A or B takes place is P ( A ∪ B )

Now , the probability of A is P ( A ) = 2/17

And , the probability of B is P ( B ) = 4/17

where P ( A ∩ B ) = 0

On simplifying the equation , we get

P ( A ∪ B ) = P ( A ) + P ( B ) - P ( A ∩ B )

So , P ( A ∪ B ) = 2/17 + 4/17

P ( A ∪ B ) = 6/17

Hence , the probability is 6/17

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The value of the points is,

(1/5, 7/5)  or (0.2, 1.4)

The given equation may be simplified as follows:

x² + 14xy + 49y² = 100

(x + 7y)(x + 7y) = 100

(x + 7y)² = 10²

x + 7y = 10

This is a straight line with the equation.

y = -(1/7)x + 10/7

The minimum distance from the origin to this line is provided by a straight line that passes through the origin and which is perpendicular to the straight line.

The slope of the perpendicular line is 7 because the product of the two slopes should be -1.

The perpendicular line is of the form

y = 7x + c.

Because the line passes through (0,0), therefore c = 0.

The line y = 7x intercepts the original line when

y = 7x = -(1/7)x + 10/7

Therefore

7x = -(1/7)x + 10/7

Multiply through by 7.

49x = -x + 10

50x = 10

x = 1/5

y = 7x = 7/5

Hence, The minimum distance is

d = √(x² + y²)

  = √[(1/5)² + (7/5)²]

  = √2

Thus, The point is (1/5, 7/5).

So, Solution are, (1/5, 7/5)  or (0.2, 1.4)

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a) The transition matrix is  [tex]\left[\begin{array}{ccc} 0.3&0.7 \\ 0.6&0.4\end{array}\right][/tex]

b) After two years, brand A is expected to have 40.4% of the market share.

c) Brand A is expected to have 37.5% of the market share.

a) The transition matrix can be constructed using the probabilities provided in the problem. Let P be the matrix where the (i, j)-th entry represents the probability of transitioning from state i to state j. In this case, there are two states: using brand A (state 1) and not using brand A (state 2).

Using the information given in the problem, we can fill in the entries of the matrix as follows:

P = [tex]\left[\begin{array}{ccc} 0.3&0.7 \\ 0.6&0.4\end{array}\right][/tex]

b) To find the percentage of the market share that brand A gets after two years, we need to multiply the initial market share vector (40% for brand A and 60% for other brands) by the transition matrix twice:

| 0.4 0.6 | × P × P = | 0.404 0.596 |

Therefore, after two years, brand A is expected to have 40.4% of the market share.

c) To find the long-run market share for brand A, we need to find the steady-state vector of the transition matrix P. This is the vector π such that:

πP = π

and

π ₁+ π₂ = 1

where π₁ is the long-run probability of being in state 1 (using brand A) and π₂ is the long-run probability of being in state 2 (not using brand A).

Solving the equations above, we get:

π = | 0.375 0.625 |

This means that in the long run, brand A is expected to have 37.5% of the market share.

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Answer: To end up due north, the pilot will need to fly the plane 3.382 degrees east of north.

Step-by-step explanation:

don't know how to explain got it form chegg

Note that the perimeter of the parallelogram is 42

recall that opposite sides of a parallelogram are congruent always

We have to to find the distance between the  points Q(6, 6 ) and R(1, -6) using the distance formula which is

d = √[(x2 - x1) ² + (y2 - y 1)²]

where d is the distance between two points with paris   (x1 , y1)

and (x2, y2).

PS = QR = √(6-1)²  + (6+6)²

    = √5² + 12²
= √(25+144

= √(169)

= 13

PQ = SR = 8

Perimeter  = 13 + 13 + 8 + 8
Perimeter = 42.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

See attached image.

Step-by-step explanation:

T = 5200 = .2 ( E - 10600)

       5200 / .2 = E - 10600

            E = 5200/.2   + 10 600 = 36600 pounds

The solution of the matrix is [tex]4G + 2F = \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 &-36\end{bmatrix}[/tex]

A matrix is a rectangular array of numbers arranged in rows and columns. The size of a matrix is given by its dimensions, which indicate the number of rows and columns in the matrix. In this question, both matrices G and F have 4 rows and 5 columns, so we say that they are 4x5 matrices.

Scalar multiplication is performed by multiplying each element of a matrix by a scalar, which is simply a number.

Now, to find the value of 4G + 2F, we need to perform scalar multiplication on each matrix and then add the results. We get:

[tex]4G = 4 * \begin{bmatrix}8 &-5 &-8& -2 &-10 \\-6& -7 &1 &9 &2 \\4& 6 &3 &7 &5 \\-4&-3 &0 &-10 & -9\end{bmatrix}\\= \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 & -36\end{bmatrix}[/tex]

[tex]2F = 2 * \begin{bmatrix}1 &8 &-2& -5 &9 \\-9& 10 &6 &-3 &0 \\4& 5 &-4 &3 &7 \\2&-10 &-6 &-1 & -8\end{bmatrix}= \begin{bmatrix}2 &16 &-4& -10 &18 \\-18& 20 &12 &-6 &0\\8& 10 &-8 &6 &14 \\4&-20 &-12 &-2 & -16\end{bmatrix}[/tex]

Now, we can perform matrix addition on 4G and 2F to get:

[tex]4G + 2F = \begin{bmatrix}32 &-20 &-32& -8 &-40 \\-24& -28 &4 &36 &8 \\16& 24 &12 &28 &20 \\-16&-12 &0 &-40 &-36\end{bmatrix}[/tex]

Therefore, the value of 4G + 2F is the 4x5 matrix given above.

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The total amount withdrawn from the account after five years is $24,927.38.

The future value of a series of annual deposits of $2000) can be calculated using the following formula:

FV = [tex]PMT x (((1 + r)^n^-^ 1) / r)[/tex]

where FV = the future value, PMT= the payment amount =  $2000 r =  the interest rate per period = 4% n =  the number of periods = 5

the interest rate and the number of periods are the same.

[tex]FV1 = $2000 x (((1 + 0.04)^5^-^ 1) / 0.04) = $11,025.52[/tex]

This is the amount that the deposits will grow to one year after the last deposit, at an interest rate of 4% per year.

We use compound interest:

FV = [tex]PV x (1 + r)^n[/tex]

we apply this formula with a present value of $11,025.52, an interest rate of 6%, and 4 periods, we get:

FV2 = [tex]$11,025.52 * (1 + 0.06)^4 = $13,901.86[/tex]

Therefore the total amount withdrawn = FV1 + FV2 = $11,025.52 + $13,901.86 = $24,927.38

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Answer:

f'(x) = 2x, so f'(x) = f'(3) = 2(3) = 6.

Answer:

a= Acute

b=Obtuse

c= Acute

d= Right

The matrix formed by performing the row operation 2R₁ + R₂ — R₂ on M will have a new R₂ = [ -8 -1 -3]

A rectangular array of numbers or mathematical objects which are arranged in rows and columns is called a matrix. Each row of a matrix is a horizontal sequence of numbers or objects that are separated by commas and enclosed within square brackets, and it represents a vector in the row space of the matrix.

performing the row operation 2R₁ + R₂ — R₂ on M, we have;

2(-3) + (-2) = -8 {row 2 column 1}

2(-1) + 1 = -1 {row 2 column 2}

2(-4) + 5 = -3 {row 2 column 3}

Therefore, the matrix formed by performing the row operation 2R₁ + R₂ — R₂ on M will have a new R₂ = [ -8 -1 -3]

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Answer: 134.

Step-by-step explanation:

First revert 67% into it's decimal form. You get 0.67. Then, multiply 0.67 by 200 because there were 200 problems.

0.67 x 200

67 x 2

134

So, christian got 134 out of 200 problems correct.

Thus, Cost of making 1  solid aluminium ball is found as  $211.95.

Something spherical is similar to a sphere in three dimensions in that it is round, or somewhat round. Even though they are never exactly round, oranges and apples are both spherical. Since an asteroid is frequently spheroidal—nearly round but lumpy—it has an approximately spherical form.

radius of solid aluminium balls r = 7.5 in.

Cost of of aluminium = $0.12 per cu. in.

Volume of sphere = 4/3 * π * r³

Volume of sphere = 4/3 * 3.14 * 7.5³

Volume of sphere = 1766.25 cu. in.

Cost of making 1  solid aluminium ball = 0.12 * 1766.25

Cost of making 1  solid aluminium ball = $211.95.

Thus, Cost of making 1  solid aluminium ball is found as  $211.95.

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