The general solution of the system of differential equation is given by x₂ = c₁(r₁[tex]e^{(r_{1} t)}[/tex]) + c₂(r₂[tex]e^{(r_{2} t)}[/tex]) where c₁ and c₂ are constants.
System of equations are ,
x₁' = X₁ + X₂ ,
x₂ = 4x₁+ x₂.
To decouple the given system of differential equations,
Eliminate one variable at a time.
Expressing x₂ in terms of x₁.
From the second equation, we have,
x₂ = 4x₁ + x₂
Rearranging this equation, we get,
⇒ x₂ - x₂ = 4x₁
⇒ 0 = 4x₁
⇒x₁ = 0
Now, let us substitute this value of x₁ into the first equation,
x₁' = x₁ + x₂
Since x₁ = 0, we have,
⇒x₁' = 0 + x₂
⇒ x₁' = x₂
Now, decoupled the system into two separate equations,
x₁' = x₂
x₂' = 4x₁ + x₂
To solve these equations, differentiate the first equation with respect to time,
x₁'' = x₂'
Substituting the value of x₂' from the second equation, we get,
x₁'' = 4x₁ + x₂
Since x₂ = x₁', we can rewrite the equation as,
⇒x₁'' = 4x₁ + x₁'
This is a second-order linear homogeneous differential equation.
Solve it by assuming a solution of the form x₁ = [tex]e^{(rt)}[/tex], where r is a constant.
Differentiating x₁ twice, we get,
x₁'' = r²[tex]e^{(rt)}[/tex]
Substituting this back into the differential equation, we have,
⇒r²[tex]e^{(rt)}[/tex] = 4[tex]e^{(rt)}[/tex] + r[tex]e^{(rt)}[/tex]
Dividing both sides by [tex]e^{(rt)}[/tex], we obtain,
⇒r² = 4 + r
Rearranging the equation, we have,
⇒r² - r - 4 = 0
To find the values of r, solve this quadratic equation.
Using the quadratic formula, we get,
r = (1 ± √(1 - 4(-4))) / 2
r = (1 ± √(1 + 16)) / 2
r = (1 ± √17) / 2
The solutions for r are,
r₁ = (1 + √17) / 2
r₂ = (1 - √17) / 2
The general solution for x₁ is given by,
x₁ = c₁[tex]e^{(r_{1} t)}[/tex] + c₂[tex]e^{(r_{2} t)}[/tex]
where c₁ and c₂ are constants.
Now, let us find x₂ using the first equation,
x₂ = x₁'
Differentiating the general solution of x₁ with respect to time, we have,
x₂ = c₁(r₁[tex]e^{(r_{1} t)}[/tex]) + c₂(r₂[tex]e^{(r_{2} t)}[/tex])
Therefore, the general solution for x₂ of the differential equation is equal to x₂ = c₁(r₁[tex]e^{(r_{1} t)}[/tex]) + c₂(r₂[tex]e^{(r_{2} t)}[/tex]) where c₁ and c₂ are constants.
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The above question is incomplete , the complete question is:
Find the general solution of the following system of differential equations by decoupling: x₁' = X₁ + X₂ , x₂ = 4x₁+ x₂.
Using standard British coins, list the number of different ways to pay 10p.
[Note: A: (2p, 1p, 1p, 2p, 1p, 2p, 1p) is not considered to be different
from B: {1p, 2p, 1p, 1p, 2p, 1p, 2p), both are just (3x2p, 4x1p}.]
The number of different combinations to pay 10p with British coins is given as follows:
6.
How to obtain the number of different combinations to pay 10 pounds with British coins?The British coins are defined as follows:
1p: one pound coin.2p: two pound coins.Hence the different combinations to pay 10p with British coins are given as follows:
10 one pound coins.1 two pound coins and 8 one pound coins.2 two pound coins and 6 one pound coins.3 two pound coins and 4 one pound coins.4 two pound coins and 2 one pound coins.5 two pound coins and 0 one pound coins.Which means that the number of combinations is of six, and the six combinations are the ones listed by the bullet points, above.
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What is an equation of the line that passes through the points (3, 3) and (-1, 7)?
Answer:
the equation of the line = y + x -3 -3 = 0
Step-by-step explanation:
Equation of the line is give n by the formula
y - y¹ = m
x - x¹
note: m is the gradient.
Point = (3, 3) and (-1, 7).
Gradient =m= y2 - y1 = 7 - 3 = 4 = -1
x2 - x1 -1 -3 -4
the equation of the line
y - y1 = m
x - x1
y - 3 = -1
x - 3
note: Perform cross multiplication also remember every whole number is over one.
=y - 3 = -1 (x - 3)
=y - 3 = - x + 3
y + x - 3 - 3 = 0.
State whether a dashed line or solid line would represent the graph of each inequality.
Part A: y > 2x − 9
b- y − 6 ≤ x
In part A; The graph of inequality y > 2x - 9 shows the dashed line.
And, In part B; The graph of inequality y - 6 ≤ x shown the solid line.
What is Inequality?A relation by which we can compare two or more mathematical expression is called an inequality.
Given that;
The inequality are,
Part A: y > 2x − 9
Part b; y − 6 ≤ x
Now,
In part A; The inequality '<' used which represent the dashed line
And, In part B; The inequality ≤ used which represent the solid line.
Hence, The graph of inequality y > 2x - 9 shows the dashed line.
And, The graph of inequality y - 6 ≤ x shown the solid line.
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Identify the perimeter and area of an equilateral triangle with height 11 cm. Give your answer in simplest radical form.
The perimeter of an equilateral triangle is equal to three times the length of the side. Perimeter = 33 cm and Area = 30.58 cm^2 (√30.58 cm^2)
Perimeter of an equilateral triangle = 3 x side
=> 3 x 11 cm = 33 cm
Area of an equilateral triangle = (√3/4) x (side^2)
=> (√3/4) x (11 cm^2)
=> (√3/4) x (121 cm^2)
=> 30.58 cm^2
=> √30.58 cm^2
The perimeter of an equilateral triangle is equal to three times the length of the side. Therefore, an equilateral triangle with a height of 11 cm will have a perimeter of 33 cm (3 x 11 cm). The area of an equilateral triangle can be calculated using the equation (√3/4) x (side^2). In this case, the area of the triangle is 30.58 cm^2 ((√3/4) x (11 cm^2)). This can be simplified to √30.58 cm^2 in its simplest radical form.
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given that 2x + 3y divideby 3x+4y ,find the ratio y : x
Although part of your question is missing, you might be referring to the question :
Given that 2x+3y / 3x+4y = 5, find the ratio of y:x
Given : 2x+3y / 3x+4y =5
To find : y : x
Solution :
(2x+3y) / (3x+4y) = 5, or
2x + 3y = 15x + 20y, or
-15x + 2x = 20y - 3y, or
-13x = 17y
Therefore, x/y = -17/13
Hence, y/x = -13/17
y : x = -13 : 17.
The answer is -13 : 17.
What is the scale factor of 10 cm 5m?
You want to build a structure with a length of 5 m, where the proportions of other dimensions are conserved. Estimate the scale factor, 10 cm / 5 m = 0.02 .
How do you figure out the scale factor?The scale factor is a number that represents the ratio of the measurements of two similar figures, such as geometric shapes or maps. To find the scale factor, you can compare corresponding side lengths or angles of the two figures. The scale factor is the ratio of the length of a side in one figure to the corresponding side in the other figure. You can also find the scale factor by taking the square root of the ratio of the areas or the ratio of the volumes of the two similar figures.
Once you have the scale factor, you can use it to find the missing measurements of a similar figure by multiplying any measurement of one figure by the scale factor. In summary, the scale factor is a numerical value that allows to relate the measurements of two similar figures.
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Given the function f(x) = 3x - 1, find x when f (x) = 20
The function f(a+1) is equal to option C. 3a + 4.
What is meant by mathematical function?A mathematical function is a rule that determines the value of the dependent variable based on the values of the independent variables. A table, a formula, or a graph are just a few examples of how a function might be expressed. A graph is considered to be a function if any vertical line formed can cross it at most one point. The graph is not a function if it contains any locations where a vertical line can pass through it twice or more. The expression y=f(x) defines the function f. You should interpret this as "y is a function of x." The input value, also known as an independent variable, is symbolized by the letter x.Given function:
f(x) = 3x + 1
We have to find f(a+1).
For this purpose, we will take x = a+1 and
substitute it in the function f(x) = 3x+1:
f(x) = 3x + 1
f(a+1) = 3(a+1) +1
f(a+1) = 3(a) + 3(1) +1
f(a+1) = 3a+3+1
f(a+1) = 3a + 4
So the function f(a+1) is equal to option C. 3a + 4.
The complete question is,
Given the function f(x) = 3x + 1, evaluate f(a + 1).
A. 3a + 1
B. a + 2
C. 3a + 4
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A man weighs 95 kg and starts an exercise and eating programme to lose weight. If he sticks to the programme, he will lose 0,4 kg per week. Calculate how much he will weigh after 36 weeks.
The weight after 36 weeks is 80.6 kg.
What is an expression?An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.
Given that, a man weighs 95 kg and starts an exercise and eating programme to lose weight.
Let the number of weeks be x.
Now, the expression is 95-0.4x
Here, x=36
95-0.4×36
= 95-14.4
= 80.6 kg
Therefore, the weight after 36 weeks is 80.6 kg.
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Two groups are hiking on a glacier. Group B starts ahead of Group A and walks . To catch up to Group B, Group A walks . : since the start of the hike Group A from the start of the trail Which equation shows Group A's distance from the start? Group B Start J
Speed = 7.7miles/2.2hours = 3.5miles/hour
Which equation shows Group A's distance from the start?In physics, the average speed is defined as the ratio of distance traveled during a certain time period.
Distance/Time = Speed
If the problem stated that the two groups’ average speeds are equal,
Mark’s Group Speed Equals Sally’s Group Speed
The problem now suggests that the time is likewise equal. That means Sally’s group took t = 4.7 – 2.5 = 2.2 hours to travel the remaining 7.7 miles.
That implies that the speed is,
7.7 miles/2.2 hours = 3.5 miles/hour.
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Marshall divides his money into four categories. He saves of his money. He gives of his money to a charity. He uses of his money to buy snacks at school, and he spends of his money on fun. What fraction of his money does he save and spend on fun?
Answer:
Marshall saves 1/4 of his money and spends 1/4 of his money on fun. Therefore, he saves and spends 1/4 + 1/4 = 1/2 or 50% of his money on those two categories.
Step-by-step explanation:
Which is a correct solution for the following system of Inequalities
HELPP DUE IN 30MIN
Answer:
I think the answer is A
1,2
Suppose that the functions and are defined for all real numbers as follows.
Write the expressions for and and evaluate .
(s+t)(x)=
(s*t)(x)=
(s-t)(-1)=
The expression (s+t)(x) is equal to the sum of the functions s(x) and t(x), the expression (s*t)(x) is equal to the product of the functions s(x) and t(x), and the expression (s-t)(-1) is equal to the difference between the functions s(-1) and t(-1). When evaluating (s+t)(-1) we get the result 0.
(s+t)(x)=s(x)+t(x)
(s*t)(x)=s(x)*t(x)
(s-t)(-1)=s(-1)-t(-1)
(s+t)(-1)=s(-1)+t(-1)=-2+2=0
(s+t)(x) = s(x)+t(x)
This means that the expression (s+t)(x) is equal to the sum of the functions s(x) and t(x).
(s*t)(x) = s(x)*t(x)
This means that the expression (s*t)(x) is equal to the product of the functions s(x) and t(x).
(s-t)(-1) = s(-1)-t(-1)
This means that the expression (s-t)(-1) is equal to the difference between the functions s(-1) and t(-1).
Finally, when we evaluate (s+t)(-1) we get s(-1)+t(-1)=-2+2=0.
The expression (s+t)(x) is equal to the sum of the functions s(x) and t(x), the expression (s*t)(x) is equal to the product of the functions s(x) and t(x), and the expression (s-t)(-1) is equal to the difference between the functions s(-1) and t(-1). When evaluating (s+t)(-1) we get the result 0.
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Solve for x= 2 11/12x-5/6(x-2/5)
The value of the equation on x = 2 is 0.5
Given Quadratic equation,
[tex]= \frac{11}{12} x - \frac{5}{6}( x - \frac{2}{5})[/tex]
Put the value of x = 2 in the above equation and we get
[tex]= \frac{11}{12} *2 - \frac{5}{6}( 2 - \frac{2}{5})[/tex]
[tex]= \frac{11}{6} - \frac{5}{6} * \frac{8}{5}[/tex]
[tex]= \frac{11}{6} - \frac{4}{3}[/tex]
[tex]= \frac{1}{2}[/tex]
= 0.5
Therefore, the value of the equation on x = 2 is 0.5
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Why is the exponent 0 1?
The exponent 0 refers to the number being raised to the power of 0.
When any number, except 0, is raised to the power of 0, the result is always 1. This is known as the "zero power property" and it is true for any non-zero number a. The mathematical statement can be represented as:
a^0 = 1
For example, 2^0 = 1, 5^0 = 1 and so on. This property holds true for any base a, including real numbers, complex numbers, and even matrices.
The exponent 1 refers to the number being raised to the power of 1. When any number is raised to the power of 1, the result is always the same as the number itself. This is known as the "identity property" and it is true for any non-zero number a. The mathematical statement can be represented as:
a^1 = a
For example, 2^1 = 2, 5^1 = 5 and so on. This property holds true for any base a, including real numbers, complex numbers, and even matrices.
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Can someone help me and show the work
ans is 90 and 30
Step-by-step explanation:
LN=90° (right angle)
MN=30°(DO THE MATH)
What are 4 ways to find a midpoint?
The four ways to find the midpoint of a line segment are using the Midpoint Formula, using the Distance Formula, using a Graph, and using a Ruler.
Using the Midpoint Formula:
This method involves using the midpoint formula, which is (x1 + x2)/2, (y1 + y2)/2. This formula gives the coordinates of the midpoint when the coordinates of the two endpoints of the line segment are known.
Using the Distance Formula:
This method involves using the distance formula to find the distance between the two endpoints of the line segment and then dividing the distance by 2. The midpoint will be located at the point that is half the distance from each endpoint.
Using a Graph:
This method involves plotting the two endpoints of the line segment on a graph and then drawing a line segment connecting the two points. The midpoint of the line segment will be located at the point where the line segment is bisected.
Using a Ruler:
This method involves measuring the distance between the two endpoints of the line segment with a ruler or measuring tape, and then marking the point that is halfway between the two endpoints. The midpoint of the line segment will be located at the point that is marked.
All of these methods can be used to find the midpoint of a line segment, it depends on what information is given and what tools are available.
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When a scaled copy is bigger than the original The scale factor is greater than 1?
The scaled copy is bigger that means each side is bigger than the original one or the factor is grater tan 1 as final length(bigger)/ original > 1
What is dilation ?In mathematics, a dilation is the enlargement or contraction of a figure by a predetermined scale factor, k, at or around a center of dilation. It is a transformation style that yields comparable figures.
Depending on the scale factor, the original figure will either enlarge or decrease in size. Scaling, also known as dilations, has no impact on how angles are measured.
The original and updated figures' angles will not change because the size has not changed but the shape has. Preimage and image are two terms that can be used to describe the first and new figures, respectively.
The original figure's vertices are identified with capital letters, such as B, to make it easier to identify between two similar figures.
The scaled copy is bigger that means each side is bigger than the original one or the factor is grater tan 1 as final length(bigger)/ original > 1
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Heidi has made a sequence of patterns using tiles. use the rule to find out how many tiles there are in the 7th pattern.
Answer:
29 is the correct answer
Would the following compound inequality be classified as an “and” inequality or an “or” inequality? Inequality: 6 > x + 1 > 3
dont guess please
The compound inequality says that x+1 is greater than 3 and less than 6.
What is inequality?When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than the relation between variables and the numbers.
Given that the compound inequality is 6 > x + 1 > 3. The compound inequallity says that the value of x +1 is greater than 3 and is less than 6.
The inequality can be written as,
6 > x + 1 > 3
x + 1 > 3
x +1 < 6
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Sydney is playing a game. She draws a fraction card labeled, Then she rolls a number cube labeled 1 through 6. sydney mutuplles the fraction b. the number rolled. If her score is greater than 1. she gets another turn. If her score is less than or equal to 1. her turn is over. What is the lowest number that Sydney can roll to get another turn? Enter the answer in the box.
Answer: To get another turn, Sydney's score must be greater than 1. Therefore, to find the lowest number that Sydney can roll to get another turn, we need to find the smallest number that, when multiplied by the fraction on the card, will give a result greater than 1.
It's not mentioned which fraction card Sydney drew, so we can assume any fraction.
Let's assume the fraction card is labeled 1/4.
To get another turn, Sydney's score must be greater than 1. Therefore, we need to find the smallest number that, when multiplied by 1/4, will give a result greater than 1.
1/4 * x >1
x > 4
so the smallest number that Sydney can roll to get another turn is 5.
It should be noted that this is assuming that the fraction card is labeled 1/4 , if the fraction card is labeled with any other fraction the answer would be different.
Step-by-step explanation:
Can you guys help me with this?
I just need to know what to do when it’s negative cause I’ve been doing positive the whole time..
The positive exponent for the given expression is 1/5.
What is a positive exponent?A positive exponent tells us how many times to multiply a base number, and a negate
ve exponent tells us how many times to divide a base number. We can rewrite negative exponents like x⁻ⁿ as 1 / xⁿ.
The given exponent is [tex]25^{-\frac{1}{2}[/tex].
Now, [tex]25^{-\frac{1}{2} }=(5^2)^{-\frac{1}{2}[/tex]
= 5⁻¹
= 1/5
Therefore, the positive exponent for the given expression is 1/5.
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PLease help me find the distance between AB BC and CA please help
The distance of AB is 42.04 units, BC is 35.77 units and CA is 34.5 units.
What is Distance?The length along a line or line segment between two points on the line or line segment.
The coordinates of A are (-18, -18)
coordinates of B are (20, 0)
coordinates of C are (-12, 16)
The formula for distance is Distance=√(x₂-x₁)²+(y₂-y₁)²
Distance AB=√(20-(-18))²+(0-18)²
=√(20+18))²+(0-18)²=√38²+18²
AB=√1444+324=√1768=42.04 units
Distance BC=√(-12-20)²+(16)²
=√(-32)²+(16)²=√1024+256
=√1280
=35.77
Distance CA=√(-12-(-18))²+(16+18)²
=√(6)²+(34)²
=√36+1156=34.5
Hence, the distance of AB is 42.04 units, BC is 35.77 units and CA is 34.5 units.
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(PLEASE HELP QUICKLY) At a particular restaurant, each pizza roll has 55 calories and each mini hot dog has 70 calories. A combination meal with pizza rolls and mini hot dogs is shown to have 910 total calories and 12 more pizza rolls than mini hot dogs. Determine the number of pizza rolls in the combination meal and the number of mini hot dogs in the combination meal.
There are____ pizza rolls and ___ mini hot dogs in the combination meal.
Answer:
There are 14 pizza rolls and 2 mini hot dogs in the combination meal.
Step-by-step explanation:
From the question, we have the following parameters
Pizza roll = 55 calories each
Mini hotdog = 70 calories each
Combination meal = 910 calories in total
Let the pizza roll be x and the mini hotdog be y
So, we have
55x + 70y = 910
x = 12 + y
Make x the subject in x = 12 + y
x = 12 + y
Substitute x = 12 + y in 55x + 70y = 910
55(12 + y) + 70y = 910
Evaluate
660 + 55y + 70y = 910
So, we have
125y = 250
This gives
y = 2
Recall that
x = 12 + y
So, we have
x = 12 + 2
x = 14
Hence, there are 14 pizza rolls and 2 mini hot dogs
Troy had a number cube with the numbers 1-6 on it.Assuming each side is equal king likeley to appear when he rolls the cube,find the following probabilities. a(rolls a 2)
Answer:
P(2)= 1/6
P(odd)= 1/2
P(factor)= 2/3
NEEDS HELP ASAP Figure B is a scaled copy of Figure A.
What is the scale factor from Figure to Figure B
Answer: 3
If you look at the left-most side on figure A, youll notice that it is 2 units long. On figure B, it is 6 units long. 2*x=6 where X is the scale factor. You'll find that X=3. YW BRO
How do you solve 3 variable equations with 3 variables?
The linear combination method will be used to resolve a system of three equations in three variables.
What is equation ?Consider the equation 3x + 5 = 14, in which the terms 3x + 5 and 14 are separated by the word "equal." The slope-intercept form, standard form, and point-slope form are the three primary formats for linear equations. A mathematical expression known as an equation has an equals sign. Many equations contain algebra. When you don't know the exact number to compute in math, algebra is used.
given
The linear combination method will be used to resolve a system of three equations in three variables.
This time, we will consider two equations at a time, eliminate one variable using the resulting equations in two variables, then solve for the third variable. with respect to this instance.
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Lily i making moothie. Each moothie ue \tfrac{1}{2} 2 1 cup of yogurt. How many cup of yogurt doe he need to make 3 moothie? Expre your anwer in implet form
Lily is making smoothie. Each smoothie 1/2 cup of yogurt. The fraction is [tex]1\frac{1}{2}[/tex] cup of yogurt does he need to make 3 smoothie.
Fraction:
Fractions represent parts of a whole. This whole thing can be a region or a collection. The word break comes from the Latin word "fraction" which means "to divide". The earliest civilization to study fractions, the Egyptians, used fractions to solve mathematical problems such as the division of food, supplies, and the lack of bullion currency.
In ancient Rome, fractions were written to represent parts of a whole using only words. In India, fractions were originally written with the numbers (numerator and denominator) on top of each other, but without lines. Arabs were the only ones who added the dash used to separate the numerator and the denomination.
According to the question:
Lily on the given conditions, formulate: 1/2 × 3
Write as a single fraction: [tex]\frac{1 * 3}{2}[/tex]
Calculate the product or quotient: 3/2
Convert to mixed fraction: [tex]1\frac{1}{2}[/tex]
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what is the opposite value of Owen's score?
Answer:
Davids Score
Step-by-step explanation:
What is a polygon with 4 called?
A 4 polygon is a two-dimensional shape with four sides and four angles.
What is polygon?Polygon is a closed figure in a plane made up of straight lines or curved lines that intersect in pairs. A polygon has at least three sides and angles. The sides can be of any length and the angles can be of any measure. Polygons are classified according to the number of sides they have, such as triangles (3 sides), quadrilaterals (4 sides), pentagons (5 sides), hexagons (6 sides), etc. Polygons are used in many fields, from mathematics to engineering.
It is a type of quadrilateral, which is a polygon with four sides. Examples of 4 polygons include squares, rectangles, rhombuses, and trapezoids. All 4 polygons have four sides of equal length, and all their interior angles add up to 360 degrees.
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A cube, of ide length 6 cm, ha the ame volume a that of a cuboid of height 3 cm and breadth 4 cm. Find the length and the urface area of the cuboid?
As per the given volume of cuboid, the length is 4 cm and the surface area of the cuboid is 5.76cm
The general formula to calculate the surface area of cuboid is written as,
SA = 2[lb + bh + hl]
where l refers the length, w refers width and h refers height.
Here we have given that a cube of side length 6 cm and has the same volume as that of a cuboid of height 3 cm and breadth 4 cm.
Now, we have to find the length and the surface area of the cuboid.
By the given value the Surface Area of Cube is calculated as.
=> 6s sq. = 6 x 4sq.
Then the value is 96 cm square
Then the Surface Area of Cuboid is calculated as,
=> 2(24+10h)=96
When we simplify the equation as
=> 24 + 10h = 48
=> 10h=24
Therefore, the value of h=2.4
Then the volume of Cuboid=6*4*2.4=5.76 cm.
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