4, 5 and 6 are quantitative variable
given that
Quantitative variablesWhen you collect quantitative data, the numbers you record represent real amounts that can be added, subtracted, divided, etc. There are two types of quantitative variables: discrete and continuous.
When you graph or plot statistical data, make sure you have quantitative data of known units. If you don’t have known units, then you won’t be able to graph it.
Variables whose values result from counting or measuring something.
Examples: height, weight, time in the 100 yard dash, number of items sold to a shopper
answer choices
1. Type of cone
2.Chocolate dipped: yes / no
3.Sprinkles: yes / no
4.Cost of chocolate dip
5.Cost of sprinkles
6.Cost of one cone
4, 5 and 6 are quantitative variable
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When the two segments are equal then the two segments are both?
When two line segments are of equal length then both the line segments are congruent to each other.
As given in the question,
Let us consider two line segments are AB and PQ.Length of the line segment AB is equal to 5 cm.And the length of the line segment PQ is equal to 2( 2.5 )cm.After simply the length of the line segment PQ is equal to 5 cm.Length of the line segment AB is equal to the length of the line segment PQ.When the length of two line segments are of equal length they are known as congruent line segments.Therefore, the two line segments with equal lengths they both are congruent to each other.
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Which of the following is a solution of the equation 2x+y =10 ?
A.x=2, y =3
B.y =0
C.x=7, y=-2
D.x= 3, y =3
The solution of the equation 2x + y = 10 is B. x = 5, y = 0.
What is an ordered pair?An ordered pair is made up of the ordinate and the abscissa of the x coordinate, with two values given in parenthesis in a certain sequence.
Pair in Order = (x, y)
x is the abscissa, the distance measure of a point from the primary axis x
y is the ordinate, the distance measure of a point from the secondary axis y
Given, A linear equation 2x + y = 10.
To find the solution we can put the solutions given in the options.
Let, x = 2, y = 3.
So, 2x + y = 10.
4 + 3 ≠ 10.
Let,
x = 7, y = - 2.
14 - 2 ≠ 10.
let,
x = 3, y = 3.
6 + 3 ≠ 10.
Let, x = 5, y = 0.
10 + 0 = 10 (satisfied).
Q.Which of the following is a solution of the equation 2x+y =10 ?
A.x=2, y =3
B.x = 5, y =0
C.x=7, y=-2
D.x= 3, y =3
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What is a line that divides a line segment into two exact half called?
Answer: the midpoint
Step-by-step explanation:
Chamberlin wants to bottle 1/8 of her apple cider. she pours the apple cider evenly among 6 bottles. what fraction of her apple cider will she put in each bottle?
On solving the provided question, we can say that fraction of her apple cider will she put in each bottle x = 1/48
what is fraction?Any number of equal portions, or fractions, can be used to represent a whole. Fractions in standard English indicate how many units of a certain size there are. 8, 3/4. A whole includes fractions. The ratio of the numerator to the denominator is how numbers are expressed in mathematics. Each of these is an integer in simple fractions. In the numerator or denominator of a complex fraction is a fraction. True fractions have numerators that are less than their denominators. A fraction is a sum that constitutes a portion of a total. By breaking the entire up into smaller bits, you can evaluate it. Half of a full number or item, for instance, is represented as 12.
Let's refer to the quantity of apple cider she will add to each bottle as:
She divides her apple cider across six bottles since, as you know, she wants to bottle 1/8 of it.
As a result, you can compose the following sentence while keeping the facts above in mind:
6x = 1/8
x = 1/6*8
x = 1/48
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The first unit of a job took 40 hours to complete. The work has a learning percentage of 88. The manager wants time estimates for units 2, 3, 4, and 5. Develop those time estimates. (Do not round intermediate calculations. Round your final answers to 2 decimal places.)
The time estimates for units 2, 3, 4, and 5, rounded to 2 decimal places, are 4.8 hours, 34.56 minutes, 4.14 minutes, and 0.48 minutes respectively.
A learning percentage is the percentage by which the time required to complete a task decreases as the number of units produced increases. In this case, the learning percentage is 88%.
To develop the time estimates for units 2, 3, 4, and 5, we can use the formula:
Time for unit n = Time for unit 1 (1 - learning percentage)⁽ⁿ⁻¹⁾
Where n is the unit number, and the learning percentage is expressed as a decimal.
For example, to find the time estimate for unit 2:
Time for unit 2 = Time for unit 1 (1 - 0.88)⁽²⁻¹⁾
= 40 (1 - 0.88)
= 40 × 0.12 = 4.8 hours
To find the time estimate for unit 3:
Time for unit 3 = Time for unit 1 (1 - 0.88)²
= 40 (1 - 0.88)²
= 40 × 0.12²
= 0.576 hours
= 34.56 minutes
To find the time estimate for unit 4:
Time for unit 4 = Time for unit 1 * (1 - 0.88)^(4-1)
= 40 * (1 - 0.88)^3
= 40 * 0.12^3
= 0.069 hours
= 4.14 minutes
To find the time estimate for unit 5:
Time for unit 5 = Time for unit 1 * (1 - 0.88)^(5-1)
= 40 * (1 - 0.88)^4
= 40 * 0.12^4
= 0.008 hours
= 0.48 minutes
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Which of the number(s) below are potential roots of the function? p(x) = x^4 + 22x^2 - 16x - 12
The numbers that are potential roots of the function given above are:
Option B-D. That is - ±1, ±2, and ±12.
It is correct to state that in mathematics, a root of an equation is a value that, when substituted into the equation, makes the equation true.
To obtain the potential roots of the expression p(x) = x⁴ + 22x² - 16x - 12:
Note that from the rational root theorem, all the possible roots of the function are in the form of a rational number.
x = ± p/q
x = ± (Factors of the constant term)/ (Factors of leading coefficient)
Thus, according to the polynomial above, we know that
The constant term is - 12The leading coefficient is 1The factors of - 12 are ±1, ±2, ±3, ±4, ±6, ±12Factors of 1 are ±1.
Thus, according to the theorem of rational root, the potential roots of the function are ±1, ±2, ±3, ±4, ±6, ±12.
Therefore, the potential roots of the function from the options listed are: are ±1, ±2, and ±12.
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Full Question:
Which of the number(s) below are potential roots of the function? p(x) = x⁴ + 22x² - 16x - 12?
± 5
±1
±2
±12
±17
PLEASE HELP ASAP!!!!!! WORTH 100 POINTS
The expression shows the differences of cubes is (5x⁷)³-(4y)³. Therefore, option C is the correct answer.
What is a perfect cube?A perfect cube of a number is a number that is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then x = y³.
A) 27x¹⁵-9y³
= 3³x¹⁵-3²y³
= (3x⁵)³-3²y³
B) 3x⁹-64y³
= 3x⁹-4³y³
= 3x⁹-(4y)³
C) 125x²¹-64y³
= 5³x²¹-4³y³
= (5x⁷)³-(4y)³
D) x⁶+27y⁹
= (x²)³+(3y)³
Therefore, option C is the correct answer.
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help me solve this answer asap please help
Answer:
11. Slope = 1, y-int = (0, -3)
Step-by-step explanation:
Move the numbers so that they math the formula y=mx+b, m being the slope, and b being the y intercept. If for example its 2y, divide the entire equation by 2. Basically you separate "y" on one side, and whatever is next to the x is your slope.
Two examples below:
a council has 6 men and 3 women. if we randomly choose two of them to co chair a committee, what is the probability these chairpersons are the same gender
There are a total of 9 women and 6 men on the council, for a total of 15 council members.
If we randomly choose 2 of them to co-chair a committee, the number of ways to choose 2 people of the same gender is:
The number of ways to choose 2 women out of the 3 women on the council is 3 choose 2 = 3.The number of ways to choose 2 men out of the 6 men on the council is 6 choose 2 = 15.
So the total number of ways to choose 2 people of the same gender is 3+15 = 18.
The total number of ways to choose 2 people without considering their gender is 15 choose 2 = 105
So the probability of choosing 2 people of the same gender is 18/105 = 6/35
So, the probability that these chairpersons are the same gender is 6/35.
A mall wading pool it’ 24 gallon of water. Water leaked out at a rate of 2 gallon every three day. At thi rate, how many day doe it take for all 24 gallon of water to leak out of the pool?
So, it would take 36 days for all 24 gallons of water to leak out of the pool at this rate
To find out how many days it takes for all 24 gallons of water to leak out of the pool, we need to determine the rate at which the water is leaking and use that to calculate the total time it takes for the pool to empty.
We know that the pool initially has 24 gallons of water and that it is leaking out at a rate of 2 gallons every 3 days. This means that the pool loses 2/3 of a gallon of water per day.
To find out how many days it will take for the pool to empty, we can divide the total amount of water in the pool by the rate at which it's leaking out.
24 gallons / (2/3 gallons/day) = 24*3/2 = 36 days
Therefore, it would take 36 days for all 24 gallons of water to leak out of the pool at this rate.
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Solve for X, Leave in simplest radical form
Therefore , the value of the given problem of trigonometry comes out to be x = 2 √3.
What is a trigonometry?Trigonometry is the branch of mathematics that studies the relationship between triangle side length The field was first developed in the Classical era, most likely in the third century BC, as a product of the fusion of geometry and astronomy research. Certain geometric equations and potential applications to calculations are covered in precise methods to mathematics. In trigonometry, there are six typical trigonometric operations. Sine, quadrature, tangent, secant, etc. are some of the names and acronyms that they go by (csc). Trigonometry is the study of triangle properties, particularly those of right triangles. However, understanding about the properties of all polygons is a necessary part of mastering geometry.
Here,
Given : Let x be the given
2 is the base
Using trigonometric ratio comes out to be :
=> Tan 30 = 2/x
=> 1/√3 = 2/x
=> x = 2 √3
Therefore , the value of the given problem of trigonometric comes out to be x = 2 √3.
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A number cube is tossed 60 times.
Outcome Frequency
1 12
2 13
3 11
4 6
5 10
6 8
Determine the experimental probability of landing on a number greater than 4.
17 over 60
18 over 60
24 over 60
42 over 60
The experimental probability of a cube tossed 60 times, landing on a number greater than 4 is the option;
18 over 60
What is the probability of an event?A probability is the fraction or proportion of the times an event will occur when an experiment or task is repeated. It is the measure of the liklyhood an event will occur
Probability compares the number of times an event occurs to the number of times all possible event outcomes occurs.
The formula for the experimental probability is presented as follows;
[tex]Probability = \frac{The \ sum \ of \ the \ times \ the \ event \ takes \ place}{The \ sum \ of \ the \ trials}[/tex]
The number of times the outcome from tossing the coin is greater than 4 obtained by adding the frequencies when for the outcomes of 5 and 6 are;
Number of outcomes greater than 4 = 10 + 8 = 18
The sum of all trials = 12 + 13 + 11 + 6 + 10 + 8 = 80
Therefore, the experimental probability = 18/60
The correct option is; 18 over 60
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Answer: B: 18/60 (18 over 60)
Step-by-step explanation:
the weight of a toy becomes 330 gm after removing 25% of its spare parts. what was the original weight of the toy.
The initial weight of the toy before removing the spare parts was found to be 440gm since the weight after 25% removal was given as 330gm
The final weight of the toy after removal of spare parts is given as 330g
Percentage weight removed in the form of spare parts from the toy= 25%
Let us consider the initial weight to be x
The equation becomes,
x – 0.25x = 330, since 25% is removed from the original weight of the toy to give us the final weight of 330gm
0.75x= 330
x = 330/0.75
x = (330*100)/75
x = 440
Therefore, initial weight of the toy was found to be 440g
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pls help
giving 100 points
The value of the expression [tex]\frac{999999999\times 999999999}{1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1}[/tex] , which involves the division of a large number is obtained by making use of a scientific calculator is; 12345678987654321
How calculators can be used to divide large numbers?Depending on the processor of the calculator, and the number of digits the calculator can handle, the exact value following the operations with very large numbers can be obtained. Graphing calculators can also be made to increase the number of digits of precession, up to 9999 digits.
The expression in the question can be presented as follows;
999999999 × 999999999 ÷ (1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1)
The value can be computed using a calculator;
999999999 × 999999999 = 999999998000000001
(1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) = 81
999999999 × 999999999 ÷ (1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1) = 999999998000000001 ÷ 81 = 12345678987654321
Therefore, the value of the expression is 12345678987654321Learn more about operations with large numbers here: https://brainly.com/question/2292046
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suppose that 70% of college women have been on a diet within the past 12 months. a sample survey interviews an srs of 267 college women. what is the probability that 75% or more of the women in the sample have been on a diet?
The probability that 75% or more of the women in the sample have been on a diet is 0.944.
The probability that 75% or more of the women in the sample have been on a diet can be calculated by using the binomial formula. The formula is P(X>=x) = 1 - P(X<x). The parameters in this case are as follows: n = 267, p = 0.7, and x = 201.5 (which is 75% of 267). Therefore, the probability that 75% or more of the women in the sample have been on a diet is 1 - P(X<201.5). Solving for this probability, P(X>=201.5) is 0.944. Therefore, the probability that 75% or more of the women in the sample have been on a diet is 0.944.
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a test has 75 questions worth 216 points total. the test consists of multiple-choice questions worth 2 points each and short answer questions worth 4 points each. how many of the total points on the test are from short-answer questions? do your figuring here.
There are 33 short-answer questions on the test and 33 * 4 = 132 points are from short-answer questions.
What is algebra ?
Algebra is a branch of mathematics in which letters and symbols are used to represent numbers and quantities in equations and formulas. Algebraic concepts include operations such as addition, subtraction, multiplication, and division.
We know that the test has 75 questions and it is worth 216 points total.
We know that multiple-choice questions worth 2 points each and short answer questions worth 4 points each.
To calculate the number of points from short-answer questions, we need to find out how many questions are short-answer and multiply that by 4 (points per question).
Let x be the number of short-answer questions.
We know that x + 75 - x = 75, which is the total number of questions on the test.
Also, we know that x * 4 + (75-x) * 2 = 216, which is the total number of points on the test.
Solving for x, we get:
x * 4 + 150 - 2x = 216
2x = 66
x = 33
So there are 33 short-answer questions on the test and 33 * 4 = 132 points are from short-answer questions.
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Solve the following system of equations graphically on the set of axes below.
y=-,t-4
2
y= ,r+2
3
Plot two lines by clicking the graph.
Click a line to delete it.
HELP ASAP
The given system of equations intersect in quadrant II at (-6,2)
What is linear equation ?
linear equation can be defined as the equation in which highest power is one.
Given system of equations,
y = (-1/3) x -4
y = (2/3) x + 2
In the first equation ,
y = (-1/3)x- 4
the 1st line slopes up with y intercept = -4
y = (2/3) x + 2
the 2nd line slopes down with y intercept = 2
By equating the equations , we get
(-1/3) x -4 = (2/3) x + 2
-4-2 = (2/3) x + (1/3) x
-6 = x
x = -6
By putting x = -6 in y = (-1/3) x -4
we get ,
y= (-1/3 ) -6 - 4
y = 2-4
y = -2
Hence, The given system of equations intersect in quadrant II at (-6,2).
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ummmm im confused thanks if you help
Answer:
480ft.it does not matter how much east or west they are.it asks how much"they are apart"
so it should be 480.
A surveyor wants to know the length of a tunnel built through a mountain according to his equipment he is located 340 m from the entrance of the tunnel at an angle of 58° to the perpendicular also according to his equipment he is 193 m from the other entrance of the tunnel other angle of 21° to the perpendicular based on these measurements find the length of the entire tunnel
Answer:
Length of the tunnel is 357.50 metersStep-by-step explanation:
We are given that,
The surveyor is located 340 meters from one entrance at an angle of 58°.
The surveyor is located 193 meters from one entrance at an angle of 21°.
Let the length of the tunnel = x meters
Using the Law of Cosines,
[tex]x^{2} =(340)^{2}+(193)^{2}-(2)(340)(193)cos(21+58)\\x^{2} =115600+37249-131240cos(79)\\x^{2} =115600+37249-25041.77255\\x^{2} =127807.2274\\x=\sqrt{127807.2274} \\x=357.50 meters\\[/tex]
Hence, the length of the tunnel is 357.50 metersHolly's garden is divided into 5 equal sections. She will fence the garden into 3 areas by grouping some equal sections together. What part of the garden would each fenced area be? please help I forgot how to do it for my lil brother just answer E
There are two ways to fence the garden in 3 areas with some equal sections.
2/5, 2/5, 1/5 and 3/5, 1/5, 1/5.
Fraction definition
A fraction depicts a segment or component of the whole. It represents the equal parts that make up the whole. A fraction is made up of two parts: the numerator and the denominator. The number at the top is the numerator, and the number at the bottom is the denominator. The denominator specifies the total number of equal parts in the whole, whereas the numerator specifies the number of equal parts that are taken.
Given The garden is divided into 5 equal pieces, each of which is 1/5 of the total garden size.
By arranging some equal parts together, she will fence the garden into three sections. There are two ways to do this.
(1/5 +1/5) = 2/5
(1/5 +1/5) = 2/5
1/5
these are the 3 areas 2/5, 2/5, 1/5.
and
(1/5 +1/5 + 1/5) = 3/5
1/5, 1/5
these are 3 areas = 3/5, 1/5, 1/5.
Hence there are only two ways to fence the garden into three areas by grouping some equal sections.
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a bag contains 8 green marbles, 12 red marbles, and 16 blue marbles. what is the probability that a person reaches into the bag and randomly selects a green marble? enter your answer in the box as a fraction in simplest form.
On solving the provided question, we can say that the probability of both = 4/9*1/3 or 4/27
What is probability?Probability theory, a subfield of mathematics, gauges the likelihood of an occurrence or a claim being true. An event's probability is a number between 0 and 1, where approximately 0 indicates how unlikely the event is to occur and 1 indicates certainty. A probability is a numerical representation of the likelihood or likelihood that a particular event will occur. Alternative ways to express probabilities are as percentages from 0% to 100% or from 0 to 1. the percentage of occurrences in a complete set of equally likely possibilities that result in a certain occurrence compared to the total number of outcomes.
18 marbles = 8 are blue and 6 are red .
probability a blue marble = 8/18 = 4/9.
probability of getting a red marble i=6/18 = 1/3.
probability of both = 4/9*1/3 or 4/27
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AB = AC and D i the mid-point of BC. (i) State the three pair of equal part in
∆ADB and ∆ADC
It is proved that 1) AB = AC, 2) angle B = angle C fo the given triangles.
What is triangle?A triangle is a polygon with three vertices and three sides. The angles of the triangle are formed by the connection of the three sides end to end at a point. The triangle's three angles add up to 180 degrees in total.
Accprding to question:Here in Delta ADB and Delta ADC.
i) Three pair of equal parts are:
AD = AD ( common side )
BD = CD ( as d is the mid point of BC)
AB = AC (given in the question)
ii) Now,
by SSS Concurrency rule,
Delta ADB\cong \Delta ADC
iii) As both triangles are congruent to each other we can compare them and say
angle B = angle C.
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Complete question:
In figure AB=AC and D is the mid point of BC state the 3 pairs of equal parts in
Triangle ADB and triangle ADC
is ADB= ADC? give reason
is <B = <C? why?
PLEASE ANSWERRR ASAP
Determine the principal stresses and the absolute maximum shear stress.
The principal stresses are the maximum normal stresses in the x, y, and z directions.
These are typically determined using Mohr's circle. The absolute maximum shear stress is the maximum shear stress that can act on a given material, which is equal to one-half the difference between the two principal stresses.
The principal stresses are determined using Mohr's circle and the absolute maximum shear stress is equal to one-half the difference between the two principal stresses.
The principal stresses are the maximum normal stresses in the x, y, and z directions.
These are typically determined using Mohr's circle. The absolute maximum shear stress is the maximum shear stress that can act on a given material, which is equal to one-half the difference between the two principal stresses.
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Consider the diagram. Line l is a perpendicular bisector of line segment R Q. It intersects line segment R Q at point T. Line l also contains point S. Line segment R S is 3 x 2. Line segment S Q is 5 x minus 8. What is QS
A triangle is a three-edged polygon with three vertices. The length of QS is equal to 17 units.
How do triangles work?
A triangle is a polygon having three vertices and three edges. It is a fundamental geometric figure. A triangle's total angles are always equal to 180 degrees.
Given that the line segment RQ is bisected at point T by the perpendicular bisector Line l. As a result, RT = QT is the length of the two lines.
Now, in ΔRST and ΔQST,
RT ≅ QT
ST ≅ ST {Common side}
Now, since the two triangles are the right-angled triangle, therefore, the two triangles are congruent. Thus, we can write,
RS = SQ
Given that the Line segment, RS is 3x+2. Line segment SQ is 5x-8.
RS = SQ
3x + 2 = 5x - 8
2 + 8 = 5x - 3x
10 = 2x
x = 5
QS = 5x - 8
= 5(5) - 8
= 25 - 8
= 17 units
Hence, the length of QS is equal to 17 units.
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What is a zero solution?
A zero solution, also known as the trivial solution, refers to a solution of a system of equations in which all variables are equal to zero.
This type of solution occurs when the coefficients in the system of equations are set up in such a way that the only solution to the system is for all the variables to be equal to zero.
A zero solution is the answer to a system of equations where all the variables are equal to zero. This happens when the coefficients in the equations are set up such that the only solution is for all the variables to be equal to zero. This type of solution is also known as the trivial solution. A trivial solution can be thought of as a “placeholder” for when there is no other meaningful solution to a system of equations.
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You and a friend are hiking and find yourselves near the top of a waterfall. There is a path to hike down that takes 1.5 miles to reach the bottom safely and go swimming. There is also a path that is 1.2 miles down to a great place to eat lunch. Your friend wants to stay at the top of the waterfall and fly his drone overhead to take pictures from the sky. A drone is a flying robot that can be remotely controlled with computer software or technology. Draw a vertical number line to help you answer the
questions.
Part A: How far would the drone need to fly up in the air to be the opposite distance from you when you are eating lunch? (3 points)
Part B: How far would the drone need to fly up in the air to be the opposite distance from you when you are swimming? (3 points)
Part C: What does the value of zero represent in this problem? (3 points)
Part D: Use absolute value to show that if you hike down to go swimming and then hike back to the top to meet your friend that you hiked the same distance in both directions. (3 points)
Help me please
Answer:
so the square root is x so when you point the square root at the rwaterfall you waill have your
Step-by-step explanation:
the drone would need to be 1.4 miles away from your to be the opposite value. pou pou pou pou pou pou pou pou pou pou pou pou ufo p
Step-by-step explanation:
Explain how to solve 5x − 2 = 8 using the change of base formula . include the solution for x in your answer. round your answer to the nearest thousandth.
x = 2 is the solution to the equation 5x - 2 = 8.
What is the logarithm?
A logarithm is a power to which a number must be raised in order to get some other number. For example, the base ten logarithms of 100 is 2, because ten raised to the power of two is 100: log 100 = 2. because. 102 = 100.
The change of base formula for logarithms allows us to change the base of a logarithm without changing the value of the logarithm.
The formula is: logb(x) = (logc(x)) / (logc(b))
To solve 5x - 2 = 8 using the change of base formula, we will first bring the equation to one side and get 2 = 5x - 8.
Then we will add 8 to both sides of the equation to get 2 + 8 = 5x, which gives us 10 = 5x.
Finally, we will divide both sides of the equation by 5 to get x = 2.
x = 2
Therefore, x = 2 is the solution to the equation 5x - 2 = 8.
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I need help asap!! Look at attached image.
Square pyramid has triangles and square, pentagonal prism has two pentagons and five rectangles, triangular prism has nine triangles and triangular pyramid has triangles
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
Net of a square pyramid consist of triangles and square.
The net of a pentagonal prism consists of two pentagons and five rectangles.
The net of a triangular prism consists of nine triangles
The net of a triangular pyramid consists of four triangles
Hence, square pyramid has triangles and square, pentagonal prism has two pentagons and five rectangles, triangular prism has nine triangles and triangular pyramid has triangles
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a rectangular prism measures 2 yd by 10 yd by 5 yd. if the dimensions of the box were all doubled, how would the surface area of the box change?
If the dimensions of the box were all doubled, then the surface area of the box be 640 square units.
Use the concept of prism defined as:
A three-dimensional solid object called a prism has two identical ends. It consists of equal cross-sections, flat sides, and identical bases.
Without bases, the prism's faces are parallelograms or rectangles. And any n-sided polygon, whether a triangle, square, rectangle, or other, might serve as the prism's basis.
Given that,
The dimension of the rectangular prism = 2yd x 10yd x 5yd
Since we know that,
Dimension of prism = length x breadth x height
length x breadth x height = 2yd x 10yd x 5yd
Therefore,
Length = 2 yd
Breadth = 10 yd
Height = 5 yd
Now if dimension doubled the,
The dimension of the new prism = 4yd x 20yd x 10yd
Then,
Length 'l' = 4 yd
Breadth 'b'= 20 yd
Height 'h' = 10 yd
Since we know that,
The surface area of a rectangular prism is,
A=2(bl+hl+hb)
Substitute the values,
A = 2(20x4 +10x4 + 10x20)
A = 640 square units.
Hence the surface area is 640 square units.
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consider an object moving along the parametrized curve with equations: x(t)=et + e–t, y(t)=e–t where t is in the time interval [0,7 ]
seconds.
(a) The maximum speed of the object on the time interval is at time .
(b) The minimum speed of the object on the time interval is at time .
In the given time interval [0,7]:
(a) the maximum speed is 1096.63 units at t = 7 and
(b) the minimum speed is 0.91 units at time t = ln2/4.
Parametric equations of the curve
x(t) = [tex]e^t + e^{-t}[/tex]
y(t) = [tex]e^{-t}[/tex]
so, dx/dt = [tex]e^t - e^{-t}[/tex]
dy/dt = [tex]-e^{-t}[/tex]
So, speed: v(t) = [tex]\sqrt{(dx/dt)^2 + (dy/dt)^2}[/tex] = [tex]\sqrt{(e^t-e^{-t})^2 + (-e^{-t})^2}[/tex]
When v(t) is maximized or minimized, evidently v² is also maximized or minimized because speed is a positive quantity.
or (speed)² = v² = [tex](e^t-e^{-t})^2 + (-e^{-t})^2[/tex] = [tex](e^{2t} + 2e^{-2t} -2)[/tex]
Now, to get the maxima or minima of a curve we have to derivate function
d/dt(v²) = 0 or d/dt[tex](e^{2t} + 2 e^{-2t} -2)[/tex]
or [tex]2e^{2t} -4 e^{-2t} = 0[/tex] or [tex]e^{2t} = 2 e^{-2t}[/tex]
taking logs on both sides
4t lne = ln2 or 4t = ln2 or t = ln2/4
so, the only critical point is t = ln2/4 ∈ [0,7]
To calculate the maxima/minima of v, we will find functional values at the critical point and boundary values of the given interval [0,7].
At t = 0 : v(0) = [tex]\sqrt{e^0 + 2e^0 - 2} = \sqrt{1+2-2}[/tex] = 1
At t = ln2/4: v(ln2/4) = [tex]\sqrt{e^{2ln2/4} + 2e^{-2ln2/4} - 2} = \sqrt{2\sqrt{2} -2 }[/tex] = 0.91
At t = 7: v(7) = [tex]\sqrt{e^{14} + 2e^{-14} - 2}[/tex] = 1096.63
Therefore, in the given time interval [0,7]: (a) the maximum speed is 1096.63 units at t = 7 and (b) the minimum speed is 0.91 units at time t = ln2/4.
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