Working with Formulas

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An equation which Samantha can use to find the side length of the base in terms of the volume and height of her puzzle is: A. a = √(3V/h)

In Mathematics and Geometry, the volume of a square pyramid can be calculated by using the following formula:

Volume of a square pyramid, V = 1/3 × a² × h

Where:

By making "a" the subject of formula, the side length of the base of a square pyramid can be determined as follows;

V = 1/3 × a² × h

3V = a²h

a² = 3V/h

By taking the square root of both sides of the equation, we have:

a = √(3V/h)

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i) The length of the arc is approximately 73.148 meters.

ii) The area of the sector is approximately 85.584 square meters.

(i) The length of an arc can be calculated using the formula:

Arc Length = (θ/360) × 2πr

where;

θ = central angle in degrees

r = radius of the circle

π = mathematical constant approximately equal to 3.14159.

Given that the central angle is 85° and the radius is 7 meters, we can substitute these values into the formula:

Arc Length = (85/360) × 2π × 7

= (17/72) × 2 × 3.14159 × 7

≈ 1.7454 × 6.28318 × 7

≈ 73.148 meters (rounded to three decimal places.

(ii) The area of a sector can be calculated using the formula:

Sector Area = (θ/360) × πr²

Given that the central angle is 85° and the radius is 7 meters, we can substitute these values into the formula:

Sector Area = (85/360) × π × 7²

= (17/72) × 3.14159 × 7²

≈ 1.7454 × 3.14159 × 49

≈ 85.584 square meters (rounded to three decimal places)

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The correct question is:

Find:

(i) the length of an arc

(ii) the area of a sector in a circle of radius 7 meters subtended by the central angle of 85°.

The best-fit linear regression equation gives us the equation y = 0.0804x + 1.1794, where the slope is 0.0804 and the y-intercept is 1.1794.

The trendline equation will give us the equation for the linear regression line, which can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Once we have the trendline equation, we can use it to make predictions about future enrollment percentages based on the number of years after 1820.

The slope of the line tells us the rate at which the enrollment percentage is increasing or decreasing over time, and the y-intercept tells us the enrollment percentage when x is equal to zero (i.e., in 1820).

Here we have find that the best-fit linear regression equation gives us the equation

=> y = 0.0804x + 1.1794,

where the slope is 0.0804 and the y-intercept is 1.1794.

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The amount that will be in Andy's account in 2 years after the addition of interest is $112.36

Andy has $100 in an account

An interest rate of 6% is compounded annually

The amount that will be present in the account after 2 years can be calculated as follows

= 100(1+ 6/100)²

= 100(1 + 0.06)²

= 100(1.06)²

= 100(1.1236)

= 112.36

Hence the amount that will be present in the account after 2 years with the addition of interest is $112.36

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The length of OP in the given circle figure is 2.

We know that the area of a sector which obtain 'A' degree at center in a circle with radius 'r' is given by,

A = (A/360)*πr²

Here length of OP represents the radius of the circle with center O.

Let OP = R.

The area of the whole circle will be = πR²

Given that the sector which intends an angle of 72 degrees in center has area 4π/5.

According to the condition,

(72/360)* πR² = 4π/5

R²/5 = 4/5

R² = 4

R = 2 [since length cannot be negative so the negative value of square root is ignored.]

Hence the length of OP is 2.

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The probability that selected team includes more women then men is [tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

The likelihood that the selected team contains more women than males matches the probability that the team contains 5, 6, 7, or 8 women (there would therefore be 4, 3, 2, or 1 man in the squad, respectively).

The likelihood of there being 5 women on the team is equal to

[tex]\frac{(^{10} _{5} )(^{9} _{4} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

Since, we can choose 5 women out of 10 in [tex](^{10} _{5} )[/tex] ways, 4 men out of 9 in [tex](^{9} _{4} )[/tex] ways, and 9 persons out of 19 in [tex](^{19} _{9})[/tex] - [tex](^{10} _{9})[/tex] - [tex](^{9} _{9})[/tex] ways (we must reject the possibility of having all men ( [tex](^{9} _{9})[/tex] ways) or all women ( [tex](^{10} _{9})[/tex] ways) because it is a must).

Similarly, we compute the odds of having 6, 7, or 8 women on the team, and because these are disjoint alternatives, the final result is the total of their probabilities, which is

[tex]\frac{Σ^{8} _{n=5} (^{10} _{n} )(^{9} _{9-n} )}{(^{19} _{9} ) - (^{10} _{9} ) - (^{9} _{9} )}[/tex]

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For the function "f (x) = 120/x²", the domain of the function "f" is (b) All non zero real numbers.

The "Domain" of a function is the set of all possible "input-values" for which the function is defined. In this case, the function is defined as:

f (x) = 120/x²

The only restriction on the input values is that the denominator cannot be equal to zero, because division by zero is undefined.

So, the domain of the function f is all non-zero real numbers.

Therefore, the correct option is (b): All non-zero real numbers.

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The given question is incomplete, the complete question is

Given that f (x) = 120/x², What is the domain of the function f ?

(a) only positive integers,

(b) All non zero real numbers,

(c) only negative integers,

(d) All real numbers including zero.

a) We are given that f(x) = x^3 - 10x^2 + 19x + 30 and x - 6 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x - 6 to get the other factor(s):

 6 │ x^3 - 10x^2 + 19x + 30

     x^3 - 6x^2

     -------------

          -4x^2 + 19x

          -(-4x^2 + 24x)

          ---------------

                     5x + 30

                     5x - 30

                     -------

                            0

Therefore, we can write:

f(x) = (x - 6)(x^2 - 4x + 5)

The quadratic factor x^2 - 4x + 5 cannot be factored further using real numbers.

b) We are given that f(x) = x^3 + 6x^2 + 5x - 12 and x + 4 is a factor of f(x). Using long division or synthetic division, we can divide f(x) by x + 4 to get the other factor(s):

-4 │ x^3 + 6x^2 + 5x - 12

     -x^3 - 4x^2

     --------------

              2x^2 + 5x

              2x^2 + 8x

              ------------

                     -3x - 12

                     -3x - 12

                     --------

                            0

Therefore, we can write:

f(x) = (x + 4)(x^2 + 2x - 3)

The quadratic factor x^2 + 2x - 3 can be factored further using the product-sum method:

x^2 + 2x - 3 = (x + 3)(x - 1)

Therefore, we have:

f(x) = (x + 4)(x + 3)(x - 1)

Roger and Trish will have completed 11/30 of the homework if they work together for 1 hour.

First, we need to find their individual rates of work, which is the reciprocal of the time it takes for each to complete the homework:

Roger's rate: 1/6 of the homework per hour

Trish's rate: 1/5 of the homework per hour

When they work together, their combined rate of work is the sum of their individual rates:

Combined rate: 1/6 + 1/5 = 11/30 of the homework per hour

After working together for 1 hour, they will have completed:

11/30 * 1 = 11/30 of the homework

Therefore, Roger and Trish will have completed 11/30 of the homework if they work together for 1 hour.

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Alan and his mom will each have completed 10 rows after 3 hours.

An expression is a combination of numbers, variables, and/or operations (such as addition, subtraction, multiplication, division, exponents, and roots) that are grouped together to represent a value.

Let x be the number of hours it takes for Alan and his mom to have planted the same number of rows.

Alan will have planted 4 + 2x rows of carrots, and his mom will have planted 7 + x rows of tomatoes.

Setting these two expressions equal to each other gives:

4 + 2x = 7 + x

Solving for x, we get:

x = 3

Thus, after 3 hours, Alan will have planted:

4 + 2(3) = 10 rows of carrots

And his mom will have planted:

7 + (3) = 10 rows of tomatoes

Therefore, they will each have completed 10 rows of vegetables after 3 hours.

To graph these equations, we can plot them on a coordinate plane, where the x-axis represents the number of hours and the y-axis represents the number of rows planted.

Alan's equation is y = 2x + 4, and his mom's equation is y = x + 7.

The graph of these two equations intersect at the point (3, 10), which represents the moment when they have each planted 10 rows.

Therefore, Alan and his mom will each have completed 10 rows after 3 hours

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Answer:15

Step-by-step explanation:

Answer:

The statement that is best supported by the information in the box plots is:

* The 8th graders completed more community service hours than the 7th graders.

The 8th grade box plot is shifted to the right of the 7th grade box plot, which indicates that the 8th graders had a higher median number of community service hours. The 8th grade box plot also has a wider spread, which indicates that there was more variation in the number of community service hours completed by the 8th graders.

However, it is important to note that the box plots do not provide information about the total number of community service hours completed by each group of students. It is possible that some 7th graders completed more community service hours than some 8th graders.

Step-by-step explanation:

Answer:The 8th graders completed more community service hours than the 7th graders

Step-by-step explanation:

Answer:

No I don't. The editor will receive 360 undelivered mail.

Step-by-step explanation:

He sends 12000 emails, 3/100 of them come back undelivered.

Multiply to find the answer:

12000 x 3/100 = 36000/100 = 360

Simple as that; the editor will find that 360 of his sent emails will come back undelivered.

Answer:

f(1) = 1

Step-by-step explanation:

Open holes on a graph means that the function is not defined at that point, so we ignore the open hole.

At f(1), there is both an open hole at y=-2 and a closed hole at y=1

We ignore the open hole as it is not defined so the answer would be the closed hole, y = 1

So f(1) is 1

Log22/Log9 express log_9 22 in terms of common logarithms

Common logarithms, is commonly describd as base-10 logarithms.

It is a type of logarithm that involve taking the logarithm of a number with respect to the base of 10.

This means that common logarithms show how many powers of ten must be increased to get a particular number. The usual logarithm of 100, for example, is 2, because 10 raised to the power of 2 equals 100.

The answer provided is based on the full question below;

-------------express log_9 22 in terms of common logarithms

a. Log 22/9

b. Log 198

c. Log22/Log9

d. Log9/Log22

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Note that given the factors above, Orr should replace the above plant. This has to do with depreciation and book value.

We need to first compute the current total cost of keeping the old plant.

Book value = $4,500,000 - $1,000,000/year x 4 years

= $500,000

Thi smean that the current total cost of keeping the old plan tfor the reamining 10 year is

Total cost of keeping old plant = $1,000,000/year x 10 years - $500,000 + $150,000

= $9 650 000

Now lets calculate the total cost of purchasing the new plant for 10 years.

$600,000 - $150,000

= $450,000 per year.

The salvage value after 10 years is $0.

Total cost of new plant = $3,500,000 + $450,000/year x 10 years = $8,000,000

Since the cost of keeping is higher than the new one, then Orr should get a new plant.

$9 650 000 > $8,000,000

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Note that the graph of the function ((x+1)²+(y+2)²=9) is attached accordingly

A coordinate graph is one that has two axes that are perpendicular to each other. These two axes are known as the x- and y-axes. The y-axis is represented by the vertical line, while the x-axis is represented by the horizontal line. A coordinate plane, a Cartesian plane, and a Cartesian coordinate system are other names for it.

The given circle has a radius of 3 units and centre is (-1,-2).

This circle can thus be drawn by first plotting the point (-1,-2) and then with this point as centre draw a circle of radius 3.

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Full question:

graph the circle

((x+1)²+(y+2)²=9)

A. Area of the rectangle, in standard form is: 14x² + 19x - 3

B. The degree of a polynomial is: 4. It has 4 terms (4x², 3y, -6x^4y, and 4.); C ALWAYS result in another polynomial.

A polynomial is a mathematical expression consisting of variables and coefficients that involves only the operations of addition, subtraction, and multiplication of non-negative integer exponents.

Part A:

Area of the rectangle = (7x - 1)(2x + 3) = 7x(2x + 3) -1(2x + 3)

= 14x² + 21x - 2x - 3

Combine like terms:

Area of the rectangle = 14x² + 19x - 3 (standard form)

Part B: Given the expression, 4x² + 3y - 6x^4y + 4,

The degree of a polynomial is the highest exponent of its variable.

Therefore, the degree is 4.

It has 4 terms, which are: 4x², 3y, -6x^4y, and 4.

Part C: When two polynomials are subtracted, multiplied, or added, it will ALWAYS result in another polynomial.

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The number of bacteria present after 13 hours is 5,306.

The number of bacteria present after 13 hours is calculated by applying half-life equation.

N = N₀ x 2^(t / d)

Where:

Since the bacteria population grew from 3,000 to 3,600 in 6 hours, the value of d is calculated as follows;

3,600 = 3,000 x 2^(6 / d)

3600/3000 = 2^(6 / d)

1.2 = 2^(6 / d)

log2 (1.2) = 6/d

0.38 = 6/d

d = 6/0.38

d = 15.8 hours

The number of bacteria after 13 hours is calculated as follows;

N = 3,000 x 2^(13 / 15.8)

N = 5,306.5 ≈ 5,306

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The complete question is below:

A culture started with 3,000 bacteria. After 6 hours, it grew to 3,600 bacteria. Predict how many bacteria will be present in 13 hours. round your answer to the nearest whole number.

Answer:

y = (3/4)x + 2

Step-by-step explanation:

A line (or linear function) can be represented by the equation y = mx + b where "m" is the slope and "b" is the y-intercept.

The y-intercept is when x = 0 or when the line crosses the y-axis. In the graph, notice that the line crosses at (0, 2). Thus, the y-intercept is at y = 2 and "b" must be "2".

Next, we have to find the slope, "m". The slope is defined as a change in "y" divided by a change in "x". Lets use the two points which are marked on the graph already, (4, 5) and (0, 2). We first subtract the y-values, and then divide that by the difference between corresponding x-values:

(5 - 2)/(4 - 0) = 3/4                           The slope is 3/4, so m = 3/4

Now, plug "m" and "b" into the equation:

y = (3/4)x + 2

Answer:

y=(3/4)x+2

Step-by-step explanation:

So first you need to find the slope (3/4). To find that you can use the rise over run method. In this case they have already chosen two points on the line, but you could use any exact points. Count how many up from 2 to 5 (in green), which is 3 then how many across (in red) which is 4. Then put the rise (3) over the run (4) so 3/4.

Then b (2) is equal to y in the y intercept which is (0,2)

Answer:

x ≥ -5

Step-by-step explanation:

Since we see the dot is filled in, we know the sign is either ≤ or ≥. Then, we can see that x is greater than -5 as the arrow is pointing in the positive direction (to the right). Therefore, it would be x ≥ -5

The balanced equation of the hanger is 3z = 45 and the solution is z = 15

From the question, we have the following parameters that can be used in our computation:

The hanger image

On the hanger image, we have the following

One side = z + z + z

The other side = 45

The expressions on either sides must be equal to have a balanced equation

So, we have

z + z + z = 45

Evaluate the like terms

3z = 45

Divide both sides of the equation by 3

z = 15

Hence, the balanced equation is 3z = 45 and the solution is z = 15

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Answer:

We can calculate the slope by taking 2 points on the line. I took A(4,4) and B(-4,0), so we get m=1/2.

2. y-intercept is the point where a line crosses the y-axis which is (0,2)

3. We have f(x)=1/2x +b

By taking a particular point on the line,we get :

2=1/2(0)+b

==》 b=2

So the slope intercept form is

y=1/2x+2

One advantage of using a dot plot is that it makes it easy to identify outliers. Option A is correct.

An outlier is a data point that is significantly different from the others in the data set, either in terms of its value or its position within the range of values. In a dot plot, outliers are represented by individual dots that are far away from the other dots, either above or below them.

By looking at the dot plot, researchers can quickly identify any outliers and investigate them further to determine why they are different and whether they should be included or excluded from the analysis.

Hence, A. is the correct option.

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The grams of each food she feeds this group of rats each day is 80 grams of food A and 100 grams of food B each day.

Let us consider that the researcher feeds x grams of food A and y grams of food B each day. Then we can restructure the  two equations on the basis of  the amount of niacin and retinol present in the food
0.11x + 0.35y = 28.4 (equation 1)
190x + 95y = 37,335 (equation 2)

We can evaluate this system of equations applying the substitution or elimination method.
Let us apply substitution method:
In case of equation 1, we can evaluate concerning x:
x = (28.4 - 0.35y) / 0.11
Staging this value for x in equation 2
190[(28.4 - 0.35y) / 0.11] + 95y
= 37,335
Applying simplification and evaluating for y
y = 100
Staging this value for y in equation 1 to evaluate x

x = 80

Then, the researcher feeds 80 grams of food A and 100 grams of food B each day.
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The statement that is true about the numbers above include the following: C. (i) is rational and (ii) is rational.

In Mathematics and Geometry, there are six (6) common types of numbers and these include the following:

In Mathematics and Geometry, a rational number can be defined a type of number which comprises fractions, integers, terminating or repeating decimals such as the 2/3, 0.12345678901234567890.

In Mathematics and Geometry, an irrational number can be defined a type of number which comprises non-terminating or non-repeating decimals such as the square root of 11 or √11.

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