Based on years of weather data, the expected low temperature T (in degrees F) in Fairbanks, Alaska, can be approximated by T=36 sin [2pi/365(t-101)]+14, where t is in days and t=0 corresponds to January 1. Predict when the coldest day of the year will occur.

The trigonometric substitution to find the indefinite integral is x = 8sec(θ).

Explanation:

To find the trigonometric substitution for the given integral, follow these steps:

Step 1: we first notice that the expression inside the square root can be written as a difference of squares:

x^2 - 64 = (x^2 - 8^2)

Step 2: substitute x = 8sec(θ), which leads to the following substitutions:

x^2 = 64sec^2(θ)
x^2 - 64 = 64 tan^2(θ)

And
dx = 8sec(θ)tan(θ) dθ

Step 3: With these substitutions, the given integral can be rewritten as:

∫ x^2(x^2 - 64)^3/2 dx = ∫ (64sec^2(θ))(64tan^2(θ))^3/2 (8sec(θ)tan(θ)) dθ

Step 4: Simplifying this expression, we get:

∫ 2^18sec^3(θ)tan^4(θ) dθ

Therefore, the trigonometric substitution to find the indefinite integral is x = 8sec(θ).

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The null hypothesis (H0) states that there is no significant difference in the mean body temperature between men and women. The alternative hypothesis (H1) states that men have a higher mean body temperature than women.

Step 1: Null and Alternative Hypotheses

The null hypothesis (H0): μ1 ≤ μ2 (There is no significant difference in the mean body temperature between men and women)

The alternative hypothesis (H1): μ1 > μ2 (Men have a higher mean body temperature than women)

Step 2: Test Statistic

The test statistic for comparing the means of two independent samples with unequal variances is the t-statistic. The formula for calculating the t-statistic is:

t = (x1 - x2) / √(s1² / n1 + s2² / n2)

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Step 3: P-Value

Using the given data:

x1 = 97.72, x2 = 97.34, s1 = 0.83, s2 = 0.63, n1 = 11, n2 = 59

Plugging these values into the t-statistic formula, we get:

t = (97.72 - 97.34) / √(0.83² / 11 + 0.63² / 59)

t = 0.38 / √(0.062 + 0.0066)

t = 0.38 / √(0.0686)

Step 4: Conclusion

At a significance level of 0.05, we compare the calculated t-statistic to the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom. If the calculated t-statistic is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 5: Confidence Interval

A confidence interval can be constructed to estimate the difference between the two population means. Using the given data and assuming a 95% confidence level, the confidence interval can be calculated using the formula:

CI = (x1 - x2) ± tα/2 × √(s1² / n1 + s2² / n²)

where CI is the confidence interval, tα/2 is the critical value from the t-distribution corresponding to a 95% confidence level, and all other variables are as defined above.

Therefore, the are:

The null hypothesis states that there is no significant difference in the mean body temperature between men and women, while the alternative hypothesis states that men have a higher mean body temperature than women.

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The statement "If f(x)≤g(x) and [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]

also diverges" is true.

If f(x)≤g(x) for all x and [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then we can conclude that

[tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] also diverges.

To see why, consider the integral [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]. Since f(x) ≤ g(x) for all x,

we have:

[tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] ≤ [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]

If [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then the integral on the right-hand side is

infinite. Since [tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] is less than or equal to an infinite integral, it

must also be infinite. Therefore, [tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] also diverges.

This can be intuitively understood by considering the fact that if g(x) is bigger than f(x), then the integral of g(x) over the same interval will also be bigger than the integral of f(x). Since the integral of g(x) is infinite, the integral of f(x) must also be infinite or else it would be possible to have an integral of g(x) that is infinite while the integral of f(x) is finite, which contradicts the given condition that f(x)≤g(x) for all x.

Therefore, the statement is true.

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the final price will stay as $52

Answer:

log₂ (4x)= log₂4 + log₂x

Step-by-step explanation:

log₂ (4x)= log₂4 + log₂x illustrates the product rule for logarithmic equations.

The product rule states that logb (mn) = logb m + logb n. In this case, b is 2, m is 4, and n is x. So,

log₂ (4x) = log₂ 4 + log₂ x.

Option A is correct, the product rule  for logarithmic equations is log₂ (4x) = log₂ 4 + log₂ x

Two or more expressions with an Equal sign is called as Equation.

The logarithm is the inverse function to exponentiation.

The product rule for logarithmic equations states that the logarithm of a product of two numbers is equal to the sum of the logarithms of the individual numbers.

logab=loga + logb

log₂ (4x) = log₂ 4 + log₂ x

Therefore, the correct illustration of the product rule  for logarithmic equations is log₂ (4x) = log₂ 4 + log₂ x

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Answer: n squared plus 10 n plus 25

Step-by-step explanation:

To make the expression a perfect square:

add a term that is equal to half the coefficient of n, squared.

the coefficient of n is 10, so half of it is 5

add 5 squared, or 25, to the expression:

n squared plus 10 n plus 25

this expression can be factored into (n+5) squared, which is a perfect square.

Answer:

0.83333333....

Step-by-step explanation:

First conversation 5/6 onto division which is 5 divided by 6 which is 8.3

Answer:

7.81 miles

Step-by-step explanation:

pythagorean theorem, 6 units downwards, and 5 east, so we have to calculate the hypotenuse, or sqrt( 6^2 + 5^2) which is sqrt61 or 7.81 miles

Answer:

Leading coefficient: -22

Degree: 2

Step-by-step explanation:

The given polynomial is:

-10v-18+v²-23v²

solving like terms, we get

-22v² - 10v - 18

The leading coefficient is the coefficient of the term with the highest degree. In this case, the term with the highest degree is -22v² and its coefficient is -22. Therefore, the leading coefficient is -22.

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of v is 2, which is the degree of the polynomial. Therefore, the degree of the polynomial is 2.

The sequence converges to: lim n→[infinity] an = ln(3) = 1.0986. So the sequence converges to 1.0986.

To determine whether the sequence converges or diverges and find the limit, we'll use the properties of logarithms and the concept of limits at infinity.

Given sequence: a_n = ln(3n² + 4) - ln(n² + 4)

Using the logarithm property, ln(a) - ln(b) = ln(a/b), we can rewrite the sequence as:

a_n = ln[(3n² + 4)/(n² + 4)]

Now, we'll find the limit as n approaches infinity:

lim (n→∞) a_n = lim (n→∞) ln[(3n² + 4)/(n² + 4)]

To evaluate this limit, we can divide both the numerator and the denominator by the highest power of n, which is n^2 in this case:

lim (n→∞) ln[(3 + 4/n²)/(1 + 4/n²)]

As n approaches infinity, the terms with n² in the denominator will approach 0:

lim (n→∞) ln[(3 + 0)/(1 + 0)] = ln(3)

So, the sequence converges, and the limit is ln(3).

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The value of the function is dy/dx = f(x) = 6x²-10x-14

Differentiation is an element of personalized learning which involves changing the instructional approach to meet the diverse needs of students. It can involve designing and delivering instruction using an assortment of teaching styles and giving students options for taking in information and making sense of ideas.

the given function f(x) 2x³ -5x² -14x + 8

F(x) =dy/dx = 2*3(x)³⁻¹ -5*2(x²⁻¹) -14(x¹⁻¹)

Therefore the derivative of the function is f(x) = 6x²-10x-14

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The required measure of the angle is m∠DCQ = 51.5° for tangent to the circle. The correct answer is option D.

Firstly, find the measure of arc ABC

As we know that the inscribed angle is half the length of the arc.

So, m∠D=(1/2)[arc ABC]

Here, m∠D=88°

Substitute and solve for arc ABC:

88°=(1/2)[arc ABC]

176° = [arc ABC]

arc ABC=176°

Now, finding the measure of arc DC:

As per the property of the complete circle,

arc ABC + arc AD + arc DC = 360°

Substitute the given values,

176° + 81° + arc DC = 360°

arc DC = 360°- 257°

arc DC = 103°

Now, Find the measure of the angle DCQ:

As we know that the inscribed angle is half the length of the arc.

So, m∠DCQ=(1/2)[arc DC]

Substitute the value of arc DC = 103°,

m∠DCQ=(1/2)[103°] = 51.5°

Thus, the required measure of the angle is m∠DCQ = 51.5°.

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The average rate of change is increasing over this interval.

To find the average rate of change of the function f(x) = x^2 + 3 from x = 0 to x = 4, we can use the formula:

average rate of change = [f(4) - f(0)] / [4 - 0]

Substituting the values of x = 0 and x = 4 into the function f(x), we get:

f(0) = 0^2 + 3 = 3

f(4) = 4^2 + 3 = 19

So, the average rate of change of the function from x = 0 to x = 4 is:

average rate of change = [f(4) - f(0)] / [4 - 0] = (19 - 3) / 4 = 4

This means that the function increases at an average rate of 4 units per unit change in x from x = 0 to x = 4.

Since the average rate of change is a constant value, the function f(x) = x^2 + 3 has a constant rate of increase from x = 0 to x = 4.

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The common ratio, |r|, is 3/7, and the geometric series is convergent with a sum of 49/4.

The given geometric series is Σ(−3)ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity. To find the common ratio, |r|, let's simplify the series.



1. Rewrite the series: Σ(−3ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity.
2. Combine the terms with the same base: Σ(−3/7)ⁿ⁻¹ * 7ⁿ⁻¹, for n = 1 to infinity.
3. Now, the common ratio, |r| = |-3/7| = 3/7.

Since |r| < 1, the geometric series is convergent.

To find the sum of the convergent series, use the formula for the sum of an infinite geometric series:

S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

4. Find the first term (n=1): a = (−3)¹⁻¹ * 7^1 = 1 * 7 = 7.
5. Use the formula: S = 7 / (1 - (3/7)) = 7 / (4/7) = 7 * (7/4) = 49/4.

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Complete question:

consider the following geometric series. [infinity] Σ(−3)ⁿ⁻¹ * 7ⁿ = 1 Find the common ratio. |r| = Determine whether the geometric series is convergent or divergent. convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

The common ratio, |r|, is 3/7, and the geometric series is convergent with a sum of 49/4.

The given geometric series is Σ(−3)ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity. To find the common ratio, |r|, let's simplify the series.



1. Rewrite the series: Σ(−3ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity.
2. Combine the terms with the same base: Σ(−3/7)ⁿ⁻¹ * 7ⁿ⁻¹, for n = 1 to infinity.
3. Now, the common ratio, |r| = |-3/7| = 3/7.

Since |r| < 1, the geometric series is convergent.

To find the sum of the convergent series, use the formula for the sum of an infinite geometric series:

S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

4. Find the first term (n=1): a = (−3)¹⁻¹ * 7^1 = 1 * 7 = 7.
5. Use the formula: S = 7 / (1 - (3/7)) = 7 / (4/7) = 7 * (7/4) = 49/4.

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Complete question:

consider the following geometric series. [infinity] Σ(−3)ⁿ⁻¹ * 7ⁿ = 1 Find the common ratio. |r| = Determine whether the geometric series is convergent or divergent. convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Answer:

The rocket's height is increasing on the interval 0<t<2.

Answer:

y(x)=7x^2+56x+115

y(x)=7(x^2+8x+115/7) ( Factor out )

y(x)=7(x^2+8x+(4)^2-1(4)^2+115/7) ( Complete the square )

y(x)=7((x+4)^2-1(4)^2+115/7) ( Use the binomial formula )

y(x)=7((x+4)^2+3/7) ( simplify )

y(x)=7*(x+4)^2+3 done!

Step-by-step explanation:

hope helps:)

If the function f(2)=25 and f' (2) = -2.5, then f(2.5) is approximately 23.75

The first-order Taylor's approximation formula, also known as the linear approximation formula, is a mathematical formula that provides an approximate value of a differentiable function f(x) near a point a. The formula is given as

f(x) ≈ f(a) + f'(a)(x - a)

where f'(a) is the derivative of f(x) at the point a. This formula is based on the tangent line to the graph of f(x) at the point (a, f(a)). The approximation becomes more accurate as x gets closer to a.

We can use the first-order Taylor's approximation formula to estimate the value of f(2.5) based on the information given

f(x) ≈ f(a) + f'(a)(x - a)

where a = 2 and x = 2.5. Plugging in the values, we get

f(2.5) ≈ f(2) + f'(2)(2.5 - 2)

f(2.5) ≈ 25 + (-2.5)(0.5)

f(2.5) ≈ 23.75

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The least number of terms needed in a Taylor polynomial to guarantee the value of ln(1.08) has an accuracy of 10⁻¹⁰ is 30. Option a is correct.

The Taylor series expansion of ln(1+x) is given by:

For ln(1.08), we have x = 0.08. Therefore, the nth term of the series is given by:

To guarantee the accuracy of ln(1.08) to 10⁻¹⁰, we need to ensure that the absolute value of the remainder term (i.e., the difference between the actual value and the value obtained using the Taylor polynomial approximation) is less than 10⁻¹⁰.

The remainder term can be bounded by the absolute value of the (n+1)th term of the series, which is:

Therefore, we need to find the smallest value of n such that:

Solving this inequality numerically, we get n > 29.82. Therefore, we need at least 30 terms in the Taylor polynomial to guarantee the accuracy of ln(1.08) to 10⁻¹⁰. Hence Option a is correct.

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The complete question is:

Estimate the least number of terms needed in a Taylor polynomial to guarantee the value of In(1.08)has accuracy of 10⁻¹⁰.

Answer: 1/63

Step-by-step explanation:

1/9 ÷ 7 can be rewritten as 1/9 x 1/7

= 1/63

Answer:

To divide a fraction by a whole number, we can flip the whole number upside down and multiply. So, 1/9 ÷ 7 is the same as 1/9 * (1/7).

To multiply fractions, we multiply the numerators and the denominators. So, 1/9 * (1/7) = (1 * 1) / (9 * 7) = 1/63.

Therefore, 1/9 ÷ 7 = 1/63.

Step-by-step explanation:

The radius of the circle is 17.78 cm.

The formula for calculating the perimeter of a circular sector with angle θ is given by

P = 2rθ

r = P / (2θ)

Substituting in the given values, we have:

r = 64 / (2 x 1.8)

r = 17.78

Therefore, the radius of the circle is 17.78 cm.

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The coordinates of N by the 270 degree rotation clockwise rule is (-7, 3)

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

N = (-3, 7)

The transfomation rule is given as

270 degree rotation rule clockwise

Mathematically, this is represented as

(x, y) = (-y, x)

Substitute the known values in the above equation, so, we have the following representation

N' = (-7, 3)

Hence, the coordinates of N after the rotation is (-7, 3)

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The line of discontinuity is x = 0 or y = 0.

We have,

To identify the line of discontinuity in the function f(x, y) = ln|x y|, we need to determine the values of x and y for which the function becomes undefined or exhibits a discontinuity.

In this case, the natural logarithm function, ln, is undefined for non-positive values.

Therefore, we need to find the values of x and y that make the expression |x y| non-positive.

The absolute value of a real number is non-positive when the number itself is zero or negative.

So, we set the expression inside the absolute value, x y, to be zero or negative:

x y ≤ 0

This inequality indicates that either x ≤ 0 and y ≥ 0, or x ≥ 0 and y ≤ 0, for the expression to be non-positive.

Hence, the line of discontinuity occurs along the line where either x ≤ 0 and y ≥ 0, or x ≥ 0 and y ≤ 0.

The equation of this line can be written as:

x ≤ 0, y ≥ 0 or x ≥ 0, y ≤ 0

This line divides the plane into two regions:

one where x ≤ 0 and y ≥ 0, and the other where x ≥ 0 and y ≤ 0.

Along this line, the function f(x, y) = ln|x y| becomes undefined or discontinuous.

Note that when x = 0 or y = 0, the function f(x, y) = ln|x y| is also undefined, but these points do not form a continuous line.

Thus,

The line of discontinuity is x = 0 or y = 0.

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The measure of angle YXZ is 9.

We are given that;

XZ= x+54

YZ= x+108

Now,

By the property of angle sum of circle

x+54+x+108=180

2x+162=180

Solving the equation

2x=180-162

2x=18

x=9

Therefore, by the angle property the answer will be 9.

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The measure of angle YXZ is 9.

We are given that;

XZ= x+54

YZ= x+108

Now,

By the property of angle sum of circle

x+54+x+108=180

2x+162=180

Solving the equation

2x=180-162

2x=18

x=9

Therefore, by the angle property the answer will be 9.

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Step-by-step explanation:

a quadratic equation has 2 zeros.

luckily we got 2 points with y = 0, so these define the zero points.

a quadratic function is usually looking like

ax² + bx + c = 0

and with the zeros being the factors, we get

y = a(x - z1)(x - z2) = a(x + 4)(x - 2) =

= a(x² - 2x + 4x - 8) = a(x² + 2x - 8)

to get "a" we use the third point.

-2 = a((-2)² + 2×-2 - 8) = a(4 - 4 - 8) = -8a

a = -2/-8 = 1/4

and the equation is

y = (1/4)x² + (1/2)x - 8/4 = (1/4)x² + (1/2)x - 2

Answer: no

Step-by-step explanation: if we'd substitute the numbers, it'd look like this 10≤-10+7  which isn't true  as "≤" this symbol means more than or equals to but -10 plus 7 is equal to 3 so it doesn't fit the inequality

B. If the right-hand-side for the Labor Time constraint decreases to 400, the dual value (shadow price) will become a less negative number, such as -4.

This is because a decrease in the available resource (Labor Time) will generally cause the shadow price to move toward a less negative value, reflecting the increased scarcity of that resource in the cost minimization problem. The correct answer is D. If the right-hand-side for the Labor Time constraint decreases to 400, it means that there is less availability of labor time, which will increase the cost of the problem. As a result, the dual value (shadow price) will become more negative, such as -8, indicating that an additional unit of labor time constraint would now cost more to relax. The allowable increase in the Labor Time constraint will decrease, while the allowable decrease will increase.

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The 95% confidence interval for the difference in means is [-0.98, 10.98], which includes 0.

To calculate the 95% confidence limits on the difference between the means (m₁ - m₂) and the difference between the standard deviations (d), we can use the following formulas:

SE(m₁ - m₂) = √[(s₁²/n₁) + (s₂²/n₂)]

where s₁ and s₂ are the sample standard deviations, n₁ and n₂ are the sample sizes, and SE represents the standard error.

95% confidence interval for (m₁ - m₂) = (x₁ - x₂) ± (t(α/2) * SE(m₁ - m₂))

where x₁ and x₂ are the sample means, t(α/2) is the t-value for the appropriate degrees of freedom and alpha level, and SE(m₁ - m₂) is the standard error.

SE(d) = √[((n₁ - 1)s₁² + (n₂ - 1)s₂²)/(n₁ + n₂ - 2)] * √[1/n₁ + 1/n₂]

where s₁ and s₂ are the sample standard deviations, n₁ and n₂ are the sample sizes, and SE represents the standard error.

95% confidence interval for d = (s₁²/s₂²) * [(n₁ + n₂ - 2)/(n₁ - 1)] * F(α/2)

where F(α/2) is the F-value for the appropriate degrees of freedom and alpha level.

Using the given data, we have:

Expectation good: n₁ = 9, x₁ = 18, s₁ = 4.38

Expectation poor: n₂ = 8, x₂ = 17.125, s₂ = 4.373

SE(m₁ - m₂) = √[(s₁²/n₁) + (s₂²/n₂)] = √[(4.38²/9) + (4.373²/8)] = 1.913

Degrees of freedom = n₁ + n₂ - 2 = 15

t(α/2) = t(0.025) = 2.131

95% confidence interval for (m₁ - m₂) = (18 - 17.125) ± (2.131 * 1.913) = (0.546, 1.429)

SE(d) = √[((n₁ - 1)s₁² + (n₂ - 1)s₂²)/(n₁ + n₂ - 2)] * √[1/n₁ + 1/n₂] = √[((8)(4.373²) + (9)(4.38²))/(17)] * √[1/8 + 1/9] = 1.322

Degrees of freedom numerator = n₁ - 1 = 8

Degrees of freedom denominator = n₂ - 1 = 7

F(α/2) = F(0.025) = 4.256

95% confidence interval for d = (4.38²/4.373²) * [(9 + 8 - 2)/(8)] * 4.256 = (0.754, 3.880)

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The requried value of x between lines m and n is 59.5.

When two lines intersect at a point, they form four angles around the intersection point. The pairs of opposite angles and sides are similar, meaning they have the proportionate measure.

As shown in the figure,
lines l,m, and n are parallel and cut by two transversal lines,
following the property of proportion of transversal line on a parallel line,
12/51 = 14/x

Simplifying the above expression,
x = 51 * [14/12]
x = 59.5

Thus, the requried value of x between lines m and n is 59.5.

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In the given problem, we can stack the sweets in six stacks, each with 24 sweets. So, there will be 24 Kaju burfies in each stack.

To stack the sweets in the least area, we want to minimize the number of stacks. To do this, we need to find the greatest common divisor (GCD) of 48 and 72, which is 24.

Therefore, we need to stack the sweets in groups of 24.

We have a total of 48 Kaju burfies, so we need to divide them into groups of 24.

48 / 24 = 2

So, we can stack the Kaju burfies in two stacks of 24 each.

We also have 72 badam becafio, which we need to stack in groups of 24.

72 / 24 = 3

So, we can stack the badam becafio in three stacks of 24 each.

Thus, we can stack the sweets in six stacks, each with 24 sweets.

So, there will be 24 Kaju burfies in each stack.

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In the given problem, we can stack the sweets in six stacks, each with 24 sweets. So, there will be 24 Kaju burfies in each stack.

To stack the sweets in the least area, we want to minimize the number of stacks. To do this, we need to find the greatest common divisor (GCD) of 48 and 72, which is 24.

Therefore, we need to stack the sweets in groups of 24.

We have a total of 48 Kaju burfies, so we need to divide them into groups of 24.

48 / 24 = 2

So, we can stack the Kaju burfies in two stacks of 24 each.

We also have 72 badam becafio, which we need to stack in groups of 24.

72 / 24 = 3

So, we can stack the badam becafio in three stacks of 24 each.

Thus, we can stack the sweets in six stacks, each with 24 sweets.

So, there will be 24 Kaju burfies in each stack.

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