What is the surface area of the rectangular prism formed by the pattern below?
Answers -
56 cm
108
136
162

To solve the exponential equation 4^(3+x) = 25 for x, we need to use logarithms. We can use any base of the logarithm to solve this equation. However, some bases may be more convenient or easier to use than others. The possible bases of the logarithm that we can use to solve this equation are:

Log base 4: If we take the logarithm of both sides of the equation with base 4, we get:

log₄(4^(3+x)) = log₄(25)

(3+x)log₄(4) = log₄(25)

3+x = log₄(25)/log₄(4)

3+x = 2.3219

x ≈ -0.6781

Log base 25: If we take the logarithm of both sides of the equation with base 25, we get:

log₂₅(4^(3+x)) = log₂₅(25)

(3+x)log₂₅(4) = 1

3+x = 1/log₂₅(4)

3+x = 1/1.3219

x ≈ -1.6781

Log base x: If we take the logarithm of both sides of the equation with base x, we get:

logₓ(4^(3+x)) = logₓ(25)

(3+x)logₓ(4) = logₓ(25)

3+x = logₓ(25)/logₓ(4)

3+x = log₄(25)/log₄(x)

x = log₄(25)/log₄(x) - 3

Log base 10: If we take the logarithm of both sides of the equation with base 10, we get:

log₁₀(4^(3+x)) = log₁₀(25)

(3+x)log₁₀(4) = log₁₀(25)

3+x = log₁₀(25)/log₁₀(4)

3+x = 1.3979

x ≈ -1.6021

Therefore, we can use any of the above logarithm bases to solve for x in the given equation.

Answer:

  C'(-5, -2)

Step-by-step explanation:

You want the coordinates of point C(4, 1) after rotation 90° CW about the origin and translation by (-6, 2).

The transformation accomplished by rotation 90° clockwise is ...

  (x, y) ⇒ (y, -x)

The translation by (-6, 2) adds those values to the coordinates:

  (x, y) ⇒ (x -6, y +2)

Combined, the transformations give you ...

  (x, y) ⇒ (y -6, -x +2)

  C(4, 1) ⇒ C'(1 -6, -4 +2) = C'(-5, -2)

The coordinates of C after the transformation(s) are (-5, -2).

The figures to complete the table, based on the pattern, would be 6. 20, 7.00.

To complete the table, you need to find the common difference between the values in B which is :

= 4. 6 - 3. 8                                         = 5. 4 - 4. 6

= 0. 80                                                   = 0. 80

The next two values would therefore be :

= 5. 4 + 0. 80                                        = 6. 20 + 0. 80

= 6. 20                                                  = 7. 00

The complete table then is :

A -  5,    6,   7,      8,       9

B- 3.8, 4.6, 5.4, 6. 20, 7. 00

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The full question is:

Complete the table based on the pattern you see:

A- 5, 6, 7, 8, 9

B- 3.8, 4.6, 5.4, ____, ____,

Step-by-step explanation:

45+6x+15=180

60+6x=180

6x=180-60

6x/6=120/6

X=20

The value of x from the same side interior angles in a parallel lines is x = 20

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line. The intersecting line is known as transversal line.

We can conclude three factors determining parallel lines ,

Alternate angles are equal

Corresponding angles are equal

Co-interior angles add up to 180°

Given data ,

Let the parallel lines be represented as A and B

Now , the measures of angles are

p = 45°

And , q = ( 6x + 15 )°

where p and q are same side interior angles

And , the same side interior angles add up to 180°

On simplifying , we get

45 + 6x + 15 = 180

6x + 60 = 180

Subtracting 60 on both sides , we get

6x = 120

Divide by 6 on both sides , we get

x = 20

Hence , the angles in parallel lines are solved and x = 20

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Julia needs to buy 42 packets of patties to make 500 hamburgers for the school function

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

Hamburger patties are sold in packets of 12. Let x represent the number of packet of patties to be bought to make 500 hamburgers. Hence:

12x = 500

x = 500/12

x = 41.67

x≅ 42

Julia needs to buy 42 packets

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"To round the decimal number 285.429 to the nearest tenth, we need to look at the digit in the hundredths place, which is 2.

If the digit in the hundredths place is 5 or greater, we round up the digit in the tenths place. If it is less than 5, we leave the digit in the tenths place as it is.

In this case, the digit in the hundredths place is 2, which is less than 5. Therefore, we leave the digit in the tenths place as it is, which is 2.

So, rounding 285.429 to the nearest tenth gives us:

285.4

Therefore, the rounded value of the decimal number 285.429 to the nearest tenth is 285.4." (ChatGPT, 2023)

Answer:

[tex]\textsf{Degree:}\quad 6[/tex]

[tex]\textsf{Leading\;coefficient:}\quad \dfrac{1}{3456}[/tex]

[tex]\textsf{Zeroes:}\quad -6, 2, 5, 8[/tex]

[tex]\textsf{Factors:}\quad (x + 6), (x - 2), (x - 5), (x - 8)[/tex]

[tex]\textsf{Factors\;with\;multiplicity:}\quad (x + 6)^3(x - 2)(x - 5)(x - 8)[/tex]

[tex]\textsf{Equation\;of\;function:}\quad f(x)=\dfrac{1}{3456}(x+6)^3(x-2)(x-5)(x-8)[/tex]

Step-by-step explanation:

The zeroes are the x-values of the points where the function crosses the x-axis, so the x-values when f(x) = 0.

Therefore, the zeroes of the given function are:

[tex]\hrulefill[/tex]

According to the factor theorem, if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x). Therefore, the factors of the function are the x-values that satisfy f(x) = 0.

Therefore, the factors of the given function are:

[tex]\hrulefill[/tex]

The multiplicity of a factor is the number of times the factor appears in the factored form of the equation of the polynomial.

If the behaviour of the x-intercept is like that of a line, i.e. the curve passes directly through the intercept, its multiplicity is one.

Therefore, the factors (x - 2), (x - 5) and (x - 8) have multiplicity one.

The behaviour of the x-intercept at x = -6 is like that of a cubic function (S-shape). There, this zero has multiplicity 3: (x + 6)³.

[tex]\hrulefill[/tex]

So far we have found the zeros, factors and their multiplicities, so we can write a factored form of the function:

[tex]\implies f(x)=a(x+6)^3(x-2)(x-5)(x-8)[/tex]

The degree of the function is the highest exponent value of the variables in the polynomial. Therefore, to find the degree of the function, simply sum the exponents of the factors:

[tex]\implies \textsf{Degree}=3 + 1 + 1 + 1 = 6[/tex]

Therefore, the degree of the function is 6.

[tex]\hrulefill[/tex]

From inspection of the given graph, the y-intercept is (0, -5).

Therefore, to find the leading coefficient (value of a), substitute (0, -5) into the equation and solve for a:

[tex]\begin{aligned}\implies f(0)=a(0+6)^3(0-2)(0-5)(0-8)&=-5\\a(216)(-2)(-5)(-8)&=-5\\-17280a&=-5\\a&=\dfrac{1}{3456}\end{aligned}[/tex]

Therefore, the leading coefficient of the function is 1/3456.

[tex]\hrulefill[/tex]

Putting everything together, the function in factored form is:

[tex]f(x)=\dfrac{1}{3456}(x+6)^3(x-2)(x-5)(x-8)[/tex]

In standard form:

[tex]f(x)=\dfrac{1}{3456}x^6+\dfrac{1}{1152}x^5-\dfrac{1}{36}x^4-\dfrac{37}{432}x^3+\dfrac{17}{24}x^2+\dfrac{13}{8}x-5[/tex]

Assuming that there are no other transactions or fees on the credit card, here's how to calculate the unpaid balance, new interest, and purchasing power on the credit card after the minimum monthly payment is made:

1.) Unpaid Balance: The unpaid balance is the amount of the credit card balance that remains after making the minimum monthly payment. In this case, the minimum monthly payment is $125, so the unpaid balance would be:

$5000 - $125 = $4875

2.) Interest Charge: The interest charge for unpaid balances is 18% per year. To calculate the monthly interest rate, divide the annual rate by 12:

18% / 12 = 1.5%

3.) To calculate the interest charge for the month, multiply the unpaid balance by the monthly interest rate:

$4875 x 1.5% = $73.13

So the new interest charge for the month is $73.13.

4.) Purchasing Power: Purchasing power refers to the amount of credit that is available on the credit card after making the minimum monthly payment and accounting for any interest charges. To calculate the purchasing power, subtract the unpaid balance and interest charge from the credit limit:

$5000 - $4875 - $73.13 = $51.87

So the purchasing power on the credit card after the minimum monthly payment is made is $51.87.

The minimum annual interest rate for Kyoko to reach her goal, using daily compounding, is 9.903%.

The interest rate is computed using an online finance calculator that sets the following parameters.

The compounding period involved is 2,555 days for 7 years, based on the assumption that there are 365 days in a year.

N (# of periods) = 2555 days (7 years x 365 days)

PV (Present Value) = $10,000

PMT (Periodic Payment) = $-0

FV (Future Value) = $-20000

Results:

I/Y = 9.903% if interest compound 365 times per year (APR)

I/Y = 10.409% if interest compound once per year (APY)

I/period = 0.027% interest per period

Total Interest = $10,000.00

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The inequality expression g - 17 ≥ 17 when solved has a solution of g ≥ 34

Given the inequality expression

g - 17 ≥ 17

To solve the inequality g - 17 ≥ 17, we need to isolate the variable g on one side of the inequality.

We can do this by adding 17 to both sides of the inequality:

g - 17 + 17 ≥ 17 + 17

Simplifying, we get:

g ≥ 34

So the solution to the inequality is all values of g that are greater than or equal to 34.

We can graph this solution on a number line by placing a closed circle at 34 and shading to the right, to indicate that any value greater than or equal to 34 satisfies the inequality.

The graph looks like this the attached figure

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Answer:

First question:

We can start by setting up the formula that relates the volume V, temperature T, and pressure P:

V = k * (T/P)

where k is a constant of proportionality. We can solve for k by plugging in the given values for V, T, and P:

70 = k * (420/18)

Simplifying the right side of the equation, we get:

70 = k * 23.3333

Dividing both sides by 23.3333, we get:

k = 3

Now we can use this value of k to find the temperature T when V is 60 cm and P is 7:

60 = 3 * (T/7)

Simplifying the equation, we get:

T/7 = 20

Multiplying both sides by 7, we get:

T = 140

Therefore, the temperature when V is 60 cm and P is 7 is 140 kelvin.

Second question: We can start by using the formula that relates the time (T), distance (D), and speed (S):

D = S * T

Since the distance between the two cities is constant, we can set the distances for the two cars to be equal:

35 * 9 = 21 * T

Simplifying the right side of the equation, we get:

315 = 21T

Dividing both sides by 21, we get:

T = 15

Therefore, the trip takes 15 hours for a car moving at 21 mph.

Hopefully this helps you! I'm sorry if it wrong. If you need more help, ask me! :]

Using the concept of percentage, we can deduce that Kareem does not have enough money with him

To solve the questions given, we need to apply the concept of percentage on some.

a) Rounding the price of the guitar to a convenient amount, we get $180.

b) 10% of the rounded price is: 0.1 x $180 = $18.

c) 1% of the rounded price is: 0.01 x $180 = $1.80.

d) To get the estimated tax, we need to multiply the rounded price by the HST rate: 0.13 x $180 = $23.40. Dividing this by $1.80, we get approximately 13.

e) The estimated tax is $23.40.

f) Adding the estimated tax to the rounded price, we get $180 + $23.40 = $203.40.

Since Kareem has $200 cash with him, he does not have enough money to buy the guitar at the estimated price of $203.40. Therefore, the answer is NO.

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Answer:

To find the cost equation, we can use the two data points given:

(100, 1400) and (400, 4100)

We can use the point-slope form of a linear equation, where the slope is the change in cost over the change in quantity:

slope = (4100 - 1400) / (400 - 100) = 2700 / 300 = 9

Using the point-slope form with the first data point:

y - 1400 = 9(x - 100)

y - 1400 = 9x - 900

y = 9x + 500

So the cost equation is y = 9x + 500.

To find the cost to manufacture 150 shoes, we can plug in x = 150 into the cost equation:

y = 9(150) + 500 = 1850

So the cost to manufacture 150 shoes is $1850.

To find the revenue function, we multiply the number of shoes sold by the price per shoe:

Revenue = price x quantity = 19x

To determine the number of items needed to break even, we need to find the quantity where revenue equals cost. Let C(x) be the cost function and R(x) be the revenue function. The break-even point occurs when:

C(x) = R(x)

9x + 500 = 19x

500 = 10x

x = 50

So the company needs to sell 50 designer shoes to break even.

Step-by-step explanation:

Answer:

Cost equation:  y = 9x + 500

It costs $1,850 to manufacture 150 shoes.

Revenue function:  R(x) = 19x

The number of items that need to be sold to break even is 50.

Step-by-step explanation:

Definition of variables

If it costs $1,400 to manufacture 100 designer shoes:

If it costs $4,100 to manufacture 400 designer shoes:

Assuming the relationship between the number of shoes and the cost to manufacture them is linear, the slope of the equation of the line the models the relationship can be found by dividing the change in y-values by the change in x-values of the two data points:

[tex]\implies \sf Slope\;(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{4100-1400}{400-100}=9[/tex]

Substitute the found slope and one of the points into the point-slope form of a linear equation to create an equation that gives the total cost to manufacture the shoes in terms of the number of shoes (x):

[tex]\implies \sf y-y_1=m(x-x_1)[/tex]

[tex]\implies \sf y-1400=9(x-100)[/tex]

[tex]\implies \sf y-1400=9x-900[/tex]

[tex]\implies \sf y=9x+500[/tex]

Therefore, the cost equation is y = 9x + 500.

To calculate the cost to manufacture 150 shoes, substitute x = 150 into the cost equation:

[tex]\begin{aligned}\implies \sf y&=\sf 9(150)+500\\&= \sf 1350+500\\&= \sf 1850\end{aligned}[/tex]

Therefore, it costs $1,850 to manufacture 150 shoes.

The revenue is the income a company generates before any expenses are subtracted. Therefore, the revenue function is simply the selling price of the item multiplied by the number of items sold.

Given the product sells for $19 per item, the revenue function is:

[tex]\implies \sf R(x)=19x[/tex]

The break even point is the point at which the total revenue equals the total cost, so there is neither profit nor loss.

To determine the number of items that should be sold to break even, equate the cost equation and the revenue function and solve for x.

[tex]\begin{aligned}\sf R(x)&=\sf y\\\implies \sf19x&=\sf 9x+500\\\sf 10x&=500\\\sf x&=50\end{aligned}[/tex]

Therefore, the number of items that need to be sold to break even is 50.

19/100 into decimals is 0,19

To write a fraction based on 10/100/1000... you must move the decimal point to the left the number of times equivalent to the number of zeros in the base.
In this case, base 100 has two zeros. Then move two squares to the left.

[tex]\begin{array}{l}\bf 19\\\sf {0}{,}\!\overset{\overset{2}{\curvearrowleft}}{1}\!\overset{\overset{1}{\curvearrowleft}}{9}\\\\\bf \dfrac{19}{100}=0{,}19\end{array}[/tex]

Did you see how easy the conversion was under these circumstances?

If you have any question about this solution, you can ask me in the comments :)

Answer:

Step-by-step explanation:

In quadrilateral ABCD, we have:

AB is parallel to DC

AD is parallel to BC

AC = 40 in (given)

BD = 50 in (given)

AB = 25 in (given)

We can use the fact that opposite sides of a parallelogram are equal in length to find the length of CD:

CD = AB = 25 in

Next, we can use the fact that the diagonals of a quadrilateral bisect each other to find the length of AO and OC:

AO = BO = BD/2 = 50/2 = 25 in

OC = OD = AC/2 = 40/2 = 20 in

Now we can use the triangle inequality to find the perimeter of triangle COD:

CO + OD + CD > OC

CO + 20 + 25 > 20

CO + 45 > 20

CO > -25

CO + OC > CO

CO + 20 > 0

CO > -20

CO + CD > OD

CO + 25 > 20

CO > -5

Therefore, the possible range of values for CO is:

-20 < CO < -5

However, since lengths cannot be negative, we can take the absolute value of CO to get:

5 < CO < 20

So the perimeter of triangle COD is:

CO + OD + CD = CO + 20 + 25 = CO + 45

Substituting the possible range of values for CO, we get:

5 + 45 < CO + 45 < 20 + 45

50 < CO + 45 < 65

So the perimeter of triangle COD is between 50 in and 65 in.

Therefore, the perimeter of triangle COD is between 50 in and 65 in, but the exact value depends on the length of CO, which is between 5 in and 20 in.

Answer:

Krish saves 9 minutes by cycling to his office.

Step-by-step explanation:

Krish used to take 30 minutes to walk to his office. After he bought a new cycle, he takes only 7/10 of the time he used to take to reach the office. Let's call the new time he takes to reach the office "t" (in minutes).

We can set up a proportion to find the value of t:

30 / 1 = t / (7/10)

To solve for t, we can cross-multiply:

30 * 7/10 = t

21 = t

So, Krish now takes 21 minutes to reach his office by cycle.

To find how many minutes he saves by cycling, we can subtract the new time from the old time:

30 - 21 = 9

Krish saves 9 minutes by cycling to his office.

Answer:

-41 is the ANS.

Step-by-step explanation:

9-25*2

9-50

-41

The inverse function of f(x) = (x + 5)² on the domain x >= -5 is given as follows:

[tex]f^{-1}(x) = \sqrt{x} - 5[/tex]

The domain of the inverse function is given as follows:

x >= 0.

The function for this problem is defined as follows:

f(x) = (x + 5)² on the domain x >= -5.

The range of the function is y >= 0, as the vertex is at (-5,0) and the function is increasing over it's domain, hence the domain of the inverse function, which is the range of the original function, is given as follows:

x >= 0.

To obtain the inverse function, wee exchange x and y in the definition of the function, and then isolate y, hence:

x = (y + 5)²

[tex]y + 5 = \sqrt{x}[/tex]

[tex]y = \sqrt{x} - 5[/tex]

[tex]f^{-1}(x) = \sqrt{x} - 5[/tex]

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Answer:     x^2-xy-2x^2y+2xy^2+2x+2y over 2x(x - y)

Step-by-step explanation:

because im that guy

Is sin (θ) = - 2/ 3, the options that are possible include:

Since sec(θ) is negative, cos(θ) is also negative (sec(θ) = 1/cos(θ)). This means θ is in the second or third quadrant, but since sin(θ) is negative, θ must be in the third quadrant. Now let's check if the given tan(θ) value is correct:

tan(θ) = sin(θ)/cos(θ)

Since we already know that cos(θ) is negative, we have cos(θ) = -(√5)/3.

Now calculate tan(θ):

tan(θ) = (-2/3) / (-(√5)/3) = 2/√5

The given values in option A are correct.

cos(θ) = -(√5)/3 and tan(θ) = 2/√5

These are the same as in option A. So, this option is also possible.

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Using square process method, we can get the values of x to be 12 and 6.

The formula a (x + m)2 + n = a (x + m)2 + n is the simplest to remember while learning how to complete the square technique.

The following formulas can be used to compute m and n in this situation: m = b/2a and n = c - (b2/4a).

From the question,

x² - 18x + 72= 0

⇒ x² - 18x = -72

Complete the square by adding the half of the square of the coefficient of x to both sides of the expression as shown:

⇒ x² - 18x + (18/2) ² = -72 + (18/2) ²

⇒ x² - 18x + 9² = -72 + 81

⇒ (x-9) ² = 9

⇒ x-9 = ± 3

So, values of x are:

x = 3 + 9

= 12

x = -3 + 9

= 6

Therefore, the values of x are 12 and 6.

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The correct option to determine the value of volume of solid figure is: B. 10 × 3 × 6 and 4 × 3 × 6.

One of the three-dimensional shapes that can be created with six pairs with parallel lines is a rectangular prism. Three-dimensional shapes in this category include cubes and cuboids. Prisms come in a variety of classifications based on the base.

Volume of rectangular prism = length * width * height

rectangular prism A:

length =  10 in.

Width =  3 in.

height = 6 in.

rectangular prism B:

length =  4 in.

Width =  3 in.

height = 6 in.

Thus,

volume of the solid = volume of rectangular prism A + volume of rectangular prism A

volume of the solid = 10*3*6 + 4*3*6

Thus, the correct option to determine the value of  volume of the solid figure is: B. 10 × 3 × 6 and 4 × 3 × 6.

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Answer:

80

Step-by-step explanation:

The variable "n" is not defined, but rather "k" is used in the summation. Assuming that "n" was meant to be "k", the expression should be:

s4 = ∑ k=1 to 4 of 2(3^(k-1))

To evaluate this expression, we need to substitute each value of k from 1 to 4 into the expression 2(3^(k-1)), and then sum up the results.

Starting with k = 1:

2(3^(1-1)) = 2(3^0) = 2(1) = 2

Moving on to k = 2:

2(3^(2-1)) = 2(3^1) = 2(3) = 6

Next, k = 3:

2(3^(3-1)) = 2(3^2) = 2(9) = 18

Finally, k = 4:

2(3^(4-1)) = 2(3^3) = 2(27) = 54

Now we add up these four results:

s4 = 2 + 6 + 18 + 54 = 80

Therefore, the value of s4 is 80.

Answer:

-sqrt(2)
-sqrt(2)
1
1

Step-by-step explanation:

Answer:

Childrens books

Step-by-step explanation:

A fraction is a fragment of a whole number, used to define parts of a whole. The whole can be a whole object, or many different objects. The number at the top of the line is called the numerator, whereas the bottom is called the denominator.

To solve for which type of book covers the most space, we need to first convert 25% from a percentage to a fraction.

A percentage is a ratio, or a number expressed in the form of a fraction of 100. Percentages are often used to express a part of a total.

If percentages are fractions of 100, then 25% as a fraction would look like this:

We can do the same to 0.3, although it isn't a percentage.

Now, we need to convert the fractions ([tex]\frac{25}{100}[/tex], [tex]\frac{30}{100}[/tex], [tex]\frac{7}{20}[/tex]) to have a common denominator.

A common denominator consists of two or more fractions that have the same denominator. This makes it easier to perform numeric equations, and to solve them.

To make [tex]\frac{7}{20}[/tex] have a denominator of 100, we can multiply the 20 by 5 to get 100.

Now the fraction is [tex]\frac{35}{100}[/tex].

The three fractions given are:

The larges given fraction is [tex]\frac{35}{100}[/tex] meaning that the children's books cover the most shelf space in the store.

Answer:

10

Step-by-step explanation:

362/6 = 60R2

461/6 = 76R5

2761/6 = 460R1

2x5x1 = 10

Answer:

1 product

Step-by-step explanation:

Given the equation that shows how the time required to complete an online shopping transaction is related to the number of products being purchased as;

t = 24p + 27

t represents the time required to complete the transaction

p is the product purchased

If it takes 51 seconds to complete a transaction, to get the amount of product purchased, we will substitute t = 51 into the expression and find p as shown;

51 = 24p + 27

Substract 27 from both sides

51-27 = 24p + 27 - 27

24 = 24p

24p= 24

p = 24/24

p = 1

Hence only 1 product is being purchased