Answer:
Larger number: 36
Smaller number: 13
A charity needs to report its typical donations received. The following is a list of the donations from one week. A histogram is provided to display the data.
10, 11, 35, 39, 40, 42, 42, 45, 49, 49, 51, 51, 52, 53, 53, 54, 56, 59
A graph titled Donations to Charity in Dollars. The x-axis is labeled 10 to 19, 20 to 29, 30 to 39, 40 to 49, and 50 to 59. The y-axis is labeled Frequency. There is a shaded bar up to 2 above 10 to 19, up to 2 above 30 to 39, up to 6 above 40 to 49, and up to 8 above 50 to 59. There is no shaded bar above 20 to 29.
Which measure of variability should the charity use to accurately represent the data? Explain your answer.
The range of 13 is the most accurate to use, since the data is skewed.
The IQR of 49 is the most accurate to use to show that they need more money.
The range of 49 is the most accurate to use to show that they have plenty of money.
The IQR of 13 is the most accurate to use, since the data is skewed.
The most appropriate measure of variability to use in this case is the IQR (Interquartile Range) of 13.
How to determine which measure of variability should the charity use to accurately represent the dataThe reason for this is that the data appears to be skewed towards the higher values, with more donations in the higher range (from 40 to 59 dollars) and fewer in the lower range (from 10 to 39 dollars).
Using the range (which is the difference between the highest and lowest values) would not accurately represent the variability of the data, as it would be heavily influenced by the outliers. On the other hand, the IQR, which is the range of the middle 50% of the data, is a more robust measure of variability that is not influenced by outliers. Therefore, the IQR of 13 is the most appropriate measure of variability to use in this case.
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If a cylinder has a height of 5 inches and a radius of 3 inches, which equation can be used to find V, the volume of the cylinder in cubic inches?
The equation to find the volume of the cylinder is: V = 45π cubic inches.
What is cubic inches?The volume of things or containers is often measured in cubic inches in the United States. It is the amount of area that a cube with one inch sides takes up. A cubic inch is about equal to 16.387 millilitres or 0.016387064 litres. The displacement of an engine—a measurement of the total amount of air and fuel the engine can compress into its cylinders—is frequently discussed in terms of cubic inches.
According to given information:The formula to find the volume of a cylinder is:
V = πr²h
Where V is the volume, r is the radius, and h is the height.
Substituting the values given in the problem, we get:
V = π(3²)(5)
V = π(9)(5)
V = 45π
Therefore, the equation to find the volume of the cylinder is:
V = 45π cubic inches.
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Leslie invests in a bank account that earns interest compounded quarterly. The expression 500(1.026)4t can be used to find the account balance in t years.
Part A
What was Leslie’s initial investment?
$
Part B
What is the quarterly interest rate the account earns?
Leslie's initial investment was $500 and the quarterly interest rate the account earns is 10.4%.
To find the initial investment and quarterly interest rate we can use the following compound formula of
A = P [tex](1 + \frac{r}{n} )^{nt}[/tex]
Given expression =[tex]500(1.026)^{4t}[/tex]
principal P = $500
rate of interest r = 1.026
A) Leslie's initial investement A = [tex]500 (1.026) ^{0}[/tex] = 500 [ initially time t = 0]
B) for finding quarterly interest,
given expression =[tex]500(1.026)^{4t}[/tex]
=[tex]500(1 + 0.026)^{4t}[/tex]
= [tex]500(1 + 0.026 * 4/4)^{4t}[/tex]
= [tex]500(1 + \frac{0.104}{4} )^{4t}[/tex]
From the above expression, we can conclude that the quarterly interest is 0.104 which is 10.4%
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State the dimensions of each matrix.
[3 -4 -9]
[2 -7 0]
Answer:
2 x 3 matrix
Step-by-step explanation:
If $4,500 is invested at an interest rate of 5.25% per year, compounded continuously, find the value of the investment after the given number of years. (Round your answers to the nearest cent.)
(a) 4 years
(b) 8 years
(c) 12 years
a) The value of the investment after 4 years is $5,544.90.
b) The value of the investment after 8 years is $6,849.42.
c) The value of the investment after 12 years is $8,453.35.
The formula for calculating the value of an investment that is compounded continuously is given by:
A = [tex]Pe^{(rt)[/tex]
where A is the final value of the investment, P is the initial investment amount, e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate expressed as a decimal, and t is the time period in years.
(a) To find the value of the investment after 4 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.05254)[/tex]
A = 4500[tex]e^{0.21[/tex]
A = 45001.2322
A = $5,544.90
(b) To find the value of the investment after 8 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.05258)[/tex]
A = 4500[tex]e^{0.42[/tex]
A = 45001.5221
A = $6,849.42
(c) To find the value of the investment after 12 years, we can substitute the given values in the formula as follows:
A = 4500[tex]e^{(0.052512)[/tex]
A = 4500[tex]e^{0.63[/tex]
A = 45001.8785
A = $8,453.35
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You are trying to determine if a router you’re thinking of purchasing will give off a strong enough wifi signal to cover the area in your house. The router you have selected will cover an area of 850 ft^2.
According to the information, the router cannot cover the kitchen area.
Does the router cover the kitchen area?To calculate if the router covers the kitchen area, we must identify the area that the router can cover. In this case, the router covers an area of 850 square feet. However, we must take into account that this area is counted around the router. So, we have to calculate the square root of 850 ft^2.
[tex]\sqrt{850}[/tex] = 29.129.1 / 2 = 14.5From the above, we can infer that the router has signal coverage of 14 meters around it, so it would only cover the living room area.
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11. Reason Inductively Complete the statement: If 8 = 2, then
m-12=. Explain.
The complete statement is, if 8 = m/4, then m - 12 = 20.
Describe Algebra?Algebra is a branch of mathematics that deals with the study of mathematical symbols and the rules for manipulating these symbols. It involves solving equations and inequalities, manipulating expressions, and working with mathematical formulas to describe and analyze relationships between variables. Algebraic equations and expressions are used to represent and solve problems in a wide range of fields, including science, engineering, economics, and finance. Algebra is an essential tool for problem-solving and critical thinking, and it provides the foundation for more advanced mathematical concepts such as calculus, statistics, and linear algebra.
The main goal of algebra is to find unknown values or variables, typically represented by letters such as x, y, or z. Algebraic equations involve a variety of mathematical operations, including addition, subtraction, multiplication, and division. These equations may also include exponents, logarithms, and other advanced mathematical concepts.
To solve the problem, we can use algebraic manipulation.
Given 8 = m/4, we can solve for m by multiplying both sides by 4:
8 × 4 = m
32 = m
Now we can substitute m = 32 into the expression m - 12 to find:
m - 12 = 32 - 12 = 20
Therefore, if 8 = m/4, then m - 12 = 20.
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The complete question is:
Complete the statement: if 8= m/4, then m-12=
Please help me!
Sloan keeps quarters and dimes in a jar, he just counted and has $23.45. If Sloane has 52 dimes in the jar, how many quarters does he have?
Answer in standard form, NEED EQUATION
x =
y =
Sloan has 73 quarters and 52 dimes kept in the jar
What is an equation?An equation is an expression that shows how numbers and variables using mathematical operators.
Let x represent the number of quarters in the jar.
1 dime = $0.10, and 1 quarter = $0.25
Hence:
He counted $23.45, of which he had 52 dimes, hence:
0.10(52) + 0.25x = 23.45
5.2 + 0.25x = 23.45
x = 73
Sloan has 73 quarters
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Help me with this question please
Answer:
A line that is perpendicular to another is one that has a slope that is the multiplicative reciprocal of the original funtion's slope and it also has the opposite sign in front. Considering a slope with a value of [tex]\dfrac{1}{7}[/tex], the reciprocal multiplicative is 7, because is a number that equals 1 if multiplied by the original value. Then, we change the sign for it, making it -7.
Thus, correct answer is a. -7.
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please find the SA and can you please explain?
Please hurry if you can.
Answer:
Rounding to the nearest hundredth, the surface area of the regular hexagonal pyramid is approximately 663.2 square centimeters.
Step by Step Explanation:
The surface area of a regular hexagonal pyramid can be calculated using the formula:
Surface area = (1/2) x Perimeter of base x Slant height + Base area
To find the perimeter of the hexagonal base, we can use the formula:
Perimeter of hexagon = 6 x Length of one side
Since the base edge length is 8 centimeters, the length of one side of the hexagon is also 8 centimeters. Therefore, the perimeter of the hexagonal base is:
Perimeter of base = 6 x Length of one side = 6 x 8 = 48 centimeters
To find the base area of the pyramid, we can use the formula:
Base area = (3/2) x (Square root of 3) x (Length of one side)^2
Substituting the values, we get:
Base area = (3/2) x (Square root of 3) x (8)^2 = 96(sqrt(3)) square centimeters
Now we can substitute the values into the surface area formula:
Surface area = (1/2) x Perimeter of base x Slant height + Base area
Surface area = (1/2) x 48 x 15 + 96(sqrt(3))
Surface area = 360 + 96(sqrt(3)) square centimeters
You’re planning to buy a house and you want to have $35,000 to put as a down payment. If you have $20,000 today and you’re able to receive 12% interest rate for investing the money, how long will it take you to have enough money for the down payment saved?
It will take approximately 6.24 years to have enough money for the down payment saved.
How to calculate how long will it take you to have enough money for the down payment savedWe can use the formula for compound interest to solve this problem:
A = P(1 + r/n)^(nt)
where:
A = the amount of money we want to have (in this case, $35,000)
P = the initial amount we have (in this case, $20,000)
r = the annual interest rate (in this case, 12%)
n = the number of times interest is compounded per year (let's assume it's compounded monthly, so n = 12)
t = the number of years we're investing for (this is what we want to find)
Substituting the values we know:
$35,000 = $20,000(1 + 0.12/12)^(12t)
Simplifying:
1.75 = (1 + 0.01)^(12t)
Taking the natural logarithm of both sides:
ln(1.75) = 12t ln(1.01)
Dividing both sides by 12 ln(1.01):
t = ln(1.75) / (12 ln(1.01))
Using a calculator:
t ≈ 6.24 years
Therefore, it will take approximately 6.24 years to have enough money for the down payment saved.
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How do I help my student understand this?
Cell 1 is in the "7th grade" row and "Pets" column. Have your student circle these labels or highlight them. Then it should be clear why choice E is the final answer.
Another way to rephrase it: the 35 people in cell 1 represent the number of 7th graders with pets.
The Celluloid Collar Corporation has $120,000 in tax loss carryforwards. The Bowstring Shirt Company, a firm in the 21% tax bracket, would be willing to pay (on a non-discounted basis) the sum of _________ for Celluloid Collar Corporation’s carryforward alone.
Group of answer choices
$18,000
$25,200
$44,100
$20,000
The Bowstring Shirt Company, a firm in the 21% tax bracket, would be willing to pay (on a non-discounted basis) the sum of $25,200 for Celluloid Collar Corporation’s carryforward alone.
What is the non-discounted sum?The value of Celluloid Collar Corporation's tax loss when it is said to be carryforward to Bowstring Shirt Company will be be calculated using the steps below
Value = Tax Loss Carryforward x Tax Rate
Value = $120,000 x 21%
= $25,200
Therefore, Bowstring Shirt Company have to pay $25,200 for Celluloid Collar Corporation's tax loss to be carryforward.
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A department manager recently launched a new initiative so the members of her department can more easily present innovative ideas they have. To get feedback on the initiative, she emails a link to all department members for an anonymous online survey.
Which statement is true?
The statement that best describes the potential bias in this scenario is option B: The survey results are most at risk for department members not being able to answer honestly.
The reason for this is that department members may be afraid to share their true thoughts and opinions, especially if they believe their ideas may be seen as too innovative or risky. They may be concerned about how their managers or colleagues will perceive them if they express ideas that go against the status quo or challenge established norms.
This type of bias is known as social desirability bias, which refers to the tendency of respondents to answer questions in a way that is socially acceptable or desirable, rather than in a way that reflects their true beliefs or behaviors.
In this case, department members may feel pressure to conform to the expectations of their superiors or colleagues, even if it means not expressing their true opinions or ideas.
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Complete Question
A department manager recently launched a new initiative so the members of her department can more easily present innovative ideas they have. To get feedback on the initiative, she emails a link to all department members for an anonymous online survey.
Which statement is true?
A. The survey results are most at risk for nonresponse bias.
B. The survey results are most at risk for department members not being able to answer honestly.
C. The survey results are not at risk of any bias.
D. The survey results are most at risk for under coverage.
question in the picturre please look at the picture
The quadratic function in standard form is f(x) = x² - 4x - 12.
What are the coefficients of the quadratic function?The coefficients of the quadratic function in standard form is calculated as follows;
The quadratic function is in the form of;
f(x) = ax² + bx + c
when x = -4, f(x) = -12
16a - 4b + c = -12
when x = -3, f(x) = -15
9a - 3b + c = -15
when x = -2, f(x) = -16
4a - 2b + c = -16
when x = -1, f(x) = -15
1a - 1b + c = -15
when x = 0, f(x) = -12
0a + 0b + c = -12
c = -12
Simplifying the equations, the value of a and b is calculated as;
16a - 4b + c = -12
9a - 3b + c = -15
4a - 2b + c = -16
a - b + c = -15
16a - 4b = 0
9a - 3b = -3
4a - 2b = -4
a - b = -3
16a = 4b
a = b/4
Substituting this expression for a into the last equation, we get:
b/4 - b = -3
b - 4b = -12
-3b = -12
b = 4
a = b/4
a = 4/4
a = 1
Therefore, the quadratic function in standard form is:
f(x) = x² - 4x - 12
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Trigonometry homework help
A. The distance she must walk to return to the starting point is approximately 1912.325 ft, or 1912 ft 3 in.
B. She is walking towards the north, the heading is N 1° 20' W.
C. Acreage = 7.55 acres (rounded to 3 decimal places).
How did we get these values?A. Convert the angles to decimal degrees:
N. 27° 40′ E. = 27.667°
N. 56° 31′ E. = 56.517°
S. 4° 26′ W. = -4.433°
Now, calculate the distances between the points:
AB = √(696.2917² + 3.5² - 2*696.2917*3.5*cos(27.667°)) = 696.2227 ft
BC = √(487.9792² + 11.75² - 2*487.9792*11.75*cos(56.517°)) = 487.9385 ft
CD = √(1691.125² + 1.5² - 2*1691.125*1.5*cos(-4.433°)) = 1691.1415 ft
B. Finding the distance from D back to A, calculate the distance and angle between AD:
AD = √(AB² + BD² - 2*AB*BD*cos(θ)) where θ = 180° - 4°26' - 27°40' = 147.9°
Substituting AB and BD:
AD = √(696.2227² + 1691.1415² - 2*696.2227*1691.1415*cos(147.9°)) = 1912.3246 ft
Therefore, the distance she must walk to return to the starting point is approximately 1912.325 ft, or 1912 ft 3 in.
C. Finding the heading she walks from D back to A, calculate the angle between AD and the north-south axis:
θ' = atan((AB*sin(27.667°) - BD*sin(θ))/AD) = atan((696.2227*sin(27.667°) - 1691.1415*sin(147.9°))/1912.3246)
θ' = -1.344°
Since she is walking towards the north, the heading is N 1° 20' W.
Lastly, to calculate the acreage of this plot of land, we use Heron's formula to determine the area of triangle ABC:
s = (AB + BC + AC)/2 = (696.2227 + 487.9385 + 1392.625)/2 = 788.8931 ft
A = √(s(s-AB)(s-BC)(s-AC)) = 328756.0207 ft²
Convert to acres:
Acreage = 328756.0207/43560 = 7.55 acres (rounded to 3 decimal places).
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Match the situation with its probability.
The given probabilities are as follows:
Spinner landing on at least one A = 7/16Spinner landing on C and D in any order = 1/8Spinner landing on two Bs = 1/16Spinner landing on C on the second spin = 1/4What are the probabilities?The probabilities are determined using the formula below:
Probability = number of expected outcome / number of total outcomesSpinner landing on at least one A:
number of at least one A = 7
Spinner landing on at least one A = 7/16
Spinner landing on C and D in any order:
number of landing on C and D in any order = 2
Spinner landing on C and D in any order = 2/16
Spinner landing on C and D in any order = 1/8
Spinner landing on two Bs:
number of landing on two Bs = 1
Spinner landing on two Bs = 1/16
Spinner landing on C on the second spin:
Number of landing on C on the second spin = 4
Spinner landing on C on the second spin = 1/4
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Suppose that the length of the three sides of a right triangle are consecutive whole numbers. What is the length of the longest leg?
The length of the longest leg in the right triangle is 5 units
What is the length of the longest leg?Let x be the length of the shortest leg. Then the other leg has length x + 1, and the hypotenuse has length x + 2.
By the Pythagorean Theorem, we have:
x^2 + (x + 1)^2 = (x + 2)^2
Set x = 3
So, we have
3^2 + (3 + 1)^2 = (3 + 2)^2
Evaluate
25 = 25
The above equation is trie
So, we have
Longest = x + 2
Longest = 3 + 2
Evaluate
Longest = 5
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help pls im failing 100 POINTS IF U HELP
Answer: 44%, 26%, less likely
Step-by-step explanation:
just do the math!!
El perímetro de un campo rectangular es 300 m . Si la longitud del campo es 88 , ¿cuál es su anchura?
Based on the above, the width of the field is 62 meters.
What is the width?Based on the question, Let's say that the width of the field is denoted with 'w'.
Note that the formula for the perimeter (P) of a rectangle is:
P = 2(l + w)
where"
l = length
w = width.
Fixing the values into the equation, it will be:
300 = 2(88 + w)
So, Divide both sides by 2:
150 = 88 + w
Subtract 88 from both sides:
w = 150 - 88
w = 62
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Se text below
The perimeter of a rectangular field is 300 m. If the length of the field is 88 , what is its width?
Las ecuaciones de la demanda y la oferta de cierto producto están dadas por espacio 4 q al cuadrado más p al cuadrado igual 1405 y p igual q más 10 donde p está en dólares por unidad y q está en unidades. Para un precio de 27 dólares por unidad, determine el gasto real del consumidor.
The actual consumer spending at a price of $27 per unit is $268.26.
How to calculate the priceSubstitute the given price into the supply equation to get the corresponding quantity supplied:
p = q + 10
27 = q + 10
q = 17
Substitute q = 17 into the demand equation to get the corresponding price:
4q² + p² = 1405
4(17)² + p² = 1405
p² = 1405 - 1156
p² = 249
p = 15.78
The actual consumer spending can be calculated by multiplying the quantity demanded by the price:
Actual consumer spending = quantity demanded x price per unit
= 17 x 15.78
= $268.26
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The demand and supply equations for a certain product are given by space 4 q squared plus p squared equals 1405 and p equals q plus 10 where p is in dollars per unit and q is in units. For a price of $27 per unit, determine the actual consumer spending.
Find the distance between the two points (3,−2) and (2,6) .
Simplify your answer, and write the exact answer in simplest radical form for an irrational answer. For example, 2–√= sqrt(2) .
Answer:
[tex] \sqrt{( {3 - 2)}^{2} + {( - 2 - 6)}^{2} } [/tex]
[tex] \sqrt{ {1}^{2} + {( - 8)}^{2} } [/tex]
[tex] \sqrt{1 + 64} = \sqrt{65} [/tex]
Use properties to rewrite the given equation. Which equations have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p? Select two options. 2.3p – 10.1 = 6.4p – 4 2.3p – 10.1 = 6.49p – 4 230p – 1010 = 650p – 400 – p 23p – 101 = 65p – 40 – p 2.3p – 14.1 = 6.4p – 4
The two equations that have the same solution as 2.3p – 10.1 = 6.5p – 4 – 0.01p include the following:
B. 2.3p – 10.1 = 6.49p – 4
C. 230p – 1010 = 650p – 400 – p
How to determine the required equations?From the information provided, we have the following equation:
2.3p – 10.1 = 6.5p – 4 – 0.01p
Next, we would simplify the given equation by collecting like terms as follows;
2.3p - 10.1 = 6.5p - 0.01p - 4 = 6.49p - 4
2.3p - 10.1 = 6.49p - 4
What is the multiplication property of equality?In Mathematics, the multiplication property of equality states that both sides of an equation will remain the same and equal, when both sides of the equations are multiplied by the same number.
By multiplying both sides of the given equation by 100, we have the following;
100 × (2.3p - 10.1) = 100 × (6.5p - 4 - 0.01p)
230p - 1010 = 650p - 400 - p
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Work the following problems using the previous tables. Choose the correct answer.
Currency on hand = 2,000 Swiss francs
Currency desired = Dutch guilders (Holland)
How many guilders based on the Wednesday rate? (Hint: Convert 2,000 Swiss francs to United States dollars. Then convert United States dollars into guilders.)
Answers:
2,533.66
2,455.75
2,465.33
The conversion of 2,000 Swiss francs to United States dollars and from United States dollars into Dutch guilders (Holland) based on Wednesday's currency exchange rate is D. ƒ4,032.
What is the exchange rate?The exchange rate represents the unit rate at which one currency is exchanged for or converted to another.
Exchange rate conversions rely on the mathematical operations of division or multiplication.
1 CHF = $1.12
2,000 CHF = $2,240 ($1.12 x 2,000 CHF)
On Wednesday April 26, 2023, $1.00 = ƒ1.80
$2,240 = ƒ4,032 ($2,240 x ƒ1.80)
Thus, we can conclude that 2,000 CHF (Swiss francs) are worth 4,032 Dutch guilders.
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Toby throws a coin 2 times.
The outcome of each throw is either Heads (H) or Tails (T).
List all the possible outcomes of the 2 throws.
Use the letters H and T in your listing (eg. HH, ...).
Answer:
HH
HT
TH
TT
Step-by-step explanation:
Answer:
TT, TH, HT, HH
Step-by-step explanation:
Wendy is customizing shirts for the debate team. She can choose from 8 styles, 7 colors, and 7 collars. How many different shirts is it possible for Wendy to customize?
Answer:
392 conbinations
Step-by-step explanation:
multiply all 3 numbers to find conbinations.
8x7x7
=56x7(or 49x8)
=392
its prob wrong cuz i guessed so yeah
A net of a rectangular prism is shown.
A net of a rectangular prism with dimensions 7 centimeters by 3 and three-fifths centimeters by 2 and two-fifths centimeters.
What is the surface area of the prism?
Answer:
The surface area of a rectangular prism is calculated by adding the areas of all six faces. The formula for the surface area of a rectangular prism is 2lw + 2lh + 2wh, where l is the length, w is the width and h is the height.
In this case, the length is 7 cm, the width is 3 and three-fifths cm (or 3.6 cm), and the height is 2 and two-fifths cm (or 2.4 cm). Plugging these values into the formula gives us a surface area of 2(7) (3.6) + 2(7) (2.4) + 2(3.6) (2.4), which simplifies to 50.4 + 33.6 + 17.28, or 101.28 square centimeters.
So, the surface area of this rectangular prism is 101.28 square centimeters.
Received message. The surface area of a rectangular prism is calculated by adding the areas of all six faces. The formula for the surface area of a rectangular prism is `2lw + 2lh + 2wh`, where `l` is the length, `w` is the width and `h` is the height. In this case, the length is 7 cm, the width is 3 and three-fifths cm (or 3.6 cm), and the height is 2 and two-fifths cm (or 2.4 cm). Plugging these values into the formula gives us a surface area of `2(7)(3.6) + 2(7)(2.4) + 2(3.6)(2.4)`, which simplifies to `50.4 + 33.6 + 17.28`, or **101.28 square centimeters**. So, the surface area of this rectangular prism is **101.28 square centimeters**.
Step-by-step explanation:
Select all the correct answers.
If the measure of angle is which statements are true?
sin (0) = -
The measure of the reference angle is 30°.
cos (0) = √3
The measure of the reference angle is 45°.
tan (0) = -√3
The measure of the reference angle is 60°.
The measures of the trigonometric relations and reference angles are solved
Given data ,
Let the measure of the angle be θ = 2π/3
Now , from the trigonometric relation ,
The tangent of the function tan ( 2π/3 ) = -√3
And , the reference angle of the θ = 2π/3 is given by A = π/3
So , A = 60°
Hence , the reference angle is 60°
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Can anyone please help me answer this question?
Find the limit of p(x)= (x^4 - x^3 - 1)/ x^2 (x + 1) as x approaches -3
Answer:
Step-by-step explanation:To find the limit of p(x) as x approaches -3, we can first simplify the expression by factoring the numerator:
p(x) = (x^4 - x^3 - 1) / x^2(x + 1)
= [(x - 1)(x^3 + x^2 - x - 1)] / [x^2(x + 1)]
Now, when x approaches -3, the denominator of the fraction becomes zero, which means we have an indeterminate form of the type 0/0. To evaluate the limit, we can use L'Hopital's rule, which states that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can take the derivative of the numerator and denominator separately and then evaluate the limit again.
Taking the derivative of the numerator and denominator, we get:
p'(x) = [(3x^2 - 2x - 1)(x^2 + 2x) - 2(x - 1)(2x + 1)] / [x^3(x + 1)^2]
Now, plugging in x = -3 into the derivative, we get:
p'(-3) = [(3(-3)^2 - 2(-3) - 1)((-3)^2 + 2(-3)) - 2((-3) - 1)(2(-3) + 1)] / [(-3)^3((-3) + 1)^2]
= [28 - 44] / [(-3)^3(-2)^2]
= -16 / 108
= -4 / 27
Since the derivative is defined and nonzero at x = -3, we can conclude that the original limit exists and is equal to the limit of the derivative, which is:
lim x->-3 p(x) = lim x->-3 [(x - 1)(x^3 + x^2 - x - 1)] / [x^2(x + 1)]
= p'(-3)
= -4 / 27
Therefore, the limit of p(x) as x approaches -3 is equal to -4/27.
Answer:
[tex]\lim_{x \to -3}p(x) =-\dfrac{107}{18}[/tex]
Step-by-step explanation:
Given the function [tex]p(x)=\dfrac{x^4-x^3-1}{x^2(x+1)}[/tex]
Let's give the expressions in the numerator and denominator their own function names so they are easy to refer to:
n, for numerator: [tex]n(x)=x^4-x^3-1[/tex]
d, for denominator: [tex]d(x)=x^2(x+1)[/tex]
So [tex]p(x)=\dfrac{n(x)}{d(x)}[/tex]
Now, we want the limit of p(x) as x goes to -3.
[tex]\lim_{x \to -3}p(x) =\lim_{x \to -3}\dfrac{n(x)}{d(x)}[/tex]
For limits of quotients, it is important to analyze the numerator and the denominator.
Take a moment to observe that inputting -3 into the denominator is defined and does not equal zero: [tex]d(-3)=(-3)^2((-3)+1)=-18\ne0[/tex]
Also, observe that inputting -3 into the numerator is defined: [tex]n(-3)=(-3)^4-(-3)^3-1=81+27-1=107[/tex]
Importantly, both functions n & d are polynomials, which are functions that are continuous over [tex]\mathbb{R}[/tex].
Since both functions n & d are continuous, both n & d are defined at [tex]x=-3[/tex], and [tex]d(-3)\ne0[/tex], then the limit of the quotient is the quotient of the limits:
[tex]\lim_{x \to -3}\dfrac{n(x)}{d(x)}=\dfrac{ \lim_{x \to -3}n(x)}{ \lim_{x \to -3}d(x)}[/tex]
From here, again, since n & d are continuous over [tex]\mathbb{R}[/tex] and defined at the limit, [tex]\lim_{x \to -3}n(x)}=n(-3)[/tex] and [tex]\lim_{x \to -3}d(x)}=d(-3)[/tex].
Therefore,
[tex]\lim_{x \to -3}p(x) =\lim_{x \to -3}\dfrac{n(x)}{d(x)}=\dfrac{ \lim_{x \to -3}n(x)}{ \lim_{x \to -3}d(x)}=\dfrac{n(-3)}{d(-3)}=\dfrac{107}{-18}=-\dfrac{107}{18}[/tex]
Which expression represents the volume of a shipping box? s(2^3) (8 + 2)3 (8x2)3 8+23
As a result, the right response is s(23), which can be written as [tex]8s^3[/tex] as to calculate a shipping box's volume is based on the box's dimensions.
what is volume ?A tri object's volume is a measurement of how much space it takes up. The usual measurement tool is a cubic unit, such as a cubic metre (m3) or cubic foot (ft3). The formula used to determine a solid item's volume depends on the shape of the object. For instance, the volume of a cube is determined by multiplying the length of a single of its sides by six, whereas the capacity of a cylinder is determined by dividing the area of its base by the diameter. Engineering, mathematics, and architecture are just a few of the disciplines where the idea of volume is crucial.
given
The formula used to calculate a shipping box's volume is based on the box's dimensions.
The box's volume can be expressed as follows if its side length is s:
[tex]V = s^3[/tex]
As a result, the right response is s(23), which can be written as [tex]8s^3[/tex] as to calculate a shipping box's volume is based on the box's dimensions.
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