help pls!! due today!​

Using the Pythagorean theorem and trig identities, it is proved sinθ + cosθ = 1.

The Pythagorean theorem states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, we can use the Pythagorean theorem to prove that sinθ+cosθ=1.

Let's take a right triangle with an angle θ and sides a, b and c, as shown in the diagram below:

By the Pythagorean theorem, we have:

a² + b² = c²

Now, we can substitute a and b with the trigonometric functions for the sides of our triangle:

sin²θ + cos²θ = 1

We can simplify this expression by using the trigonometric identity:

sin²θ + cos²θ = 1

sin²θ + cos²θ = (sinθ + cosθ)²

Substituting (sinθ + cosθ)² back into the equation, we get:

(sinθ + cosθ)² = 1

Hence, using the Pythagorean theorem and trig identities, it is proved sinθ + cosθ = 1.

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Based on a system of equations, a pound of apples costs $0.89.

A system of equations is two or more equations solved at the same time or concurrently.

A system of equations is also referred to as simultaneous equations.

                    Apples     Pears     Total Cost

Janice              2              3             $3.85

Lisa                  4              2             $4.94

Let the cost of a pound of apples = x

Let the cost of a pound of pears = y

2x + 3y = 3.85 ... Equation 1

4x + 2y = 4.94 ... Equation 2

Multiply Equation 1 by 2:

4x + 6y = 7.7 ... Equation 3

Subtract Equation 2 from Equation 3:

4x + 6y = 7.7

-

4x + 2y = 4.94

4y = 2.76

y = 0.69

Substitute y = 0.69 in Equation 1 or 2:

4x + 2y = 4.94

4x + 2(0.69) = 4.94

4x + 1.38 = 4.94

4x = 3.56

x = 0.89

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Answer: 0.71

Step-by-step explanation:

[tex]a^2+b^2=c^2\\a=b\\so\\2a^2=c^2\\c=1 here\\2a^2=1^2\\a=\sqrt{1/2} \\a=0.71[/tex]

The cost of one gallon of gas is given as follows:

$2.75.

The cost of one gallon of gas is obtained applying the proportions in the context of the problem.

A proportion is applied as the cost per gallon is obtained dividing the total cost by the number of gallons.

The parameters for this problem are given as follows:

Hence the cost of one gallon of gas is given as follows:

22/8 = $2.75.

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Using iterations, the solution to the above equation  6⁽⁻ˣ⁾ +4 = 3x -1 is Option C  = 27/16. See the graph attached.

The question requires you to state the solution of the equation. On the graph, this would be the point of intersection of both curves.

To solve for x, we'll continue using an iterative method called the fixed-point iteration ..

Rewrite the equation in the form x = g(x):

g(x) = (6⁽⁻ˣ⁾ + 5) / 3

Start with an initial guess, let's say x0 = 1.

Iterate using the formula x(n+1) = g( x(n )) until convergence, where n is the iteration number:

x (1) = g(x 0)

x (2 ) = g (x(1))

x( 3) = g(x (2))

Let's perform three iterations to approximate the solution

Iteration 1

x (1) = g( x0) = (6⁻¹ + 5) / 3

= (1/ 6 +5) / 3

= (1 /6 + 30/6) / 3

= 31/18

≈ 1.7222

Second iteration is

x(2) = g(x (1)) = ([tex]6^{1.72222}[/tex] + 5) / 3 ≈ 1.6806

Iteration 3:

x(3) = g(x (2)) = ([tex]6^-1.6806[/tex] + 5) / 3 ≈ 1.6875

After three iterations, the approximate solution to the equation 6⁽⁻ˣ⁾ + 4 = 3x - 1 is x ≈ 1.6875, which can also be expressed as the fraction 27/16.

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The Length of the ant to inches before we can compare the two lengths. We do this by dividing the numerator (3) by the denominator (32), which gives us a decimal of approximately 0.09375 inches.

To find how many times greater the length of the green anole lizard is compared to the crazy ant, we need to divide the length of the lizard by the length of the ant:

Length of lizard / Length of ant = 6 / (3/32) = 6 * (32/3) = 64

Therefore, the length of the green anole lizard is 64 times greater than the length of the crazy ant.

The correct answer is (D) 64.

It is important to pay attention to the units when working with ratios and proportions. In this case, the length of the lizard is given in inches, while the length of the ant is given in fractions of an inch. We need to convert the length of the ant to inches before we can compare the two lengths. We do this by dividing the numerator (3) by the denominator (32), which gives us a decimal of approximately 0.09375 inches.

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The square root of 107.6 is approximately 10.372

To find the square root of 107.6, we can use a calculator or use the following procedure:

Make an initial estimate:

We know that the square root of 100 is 10, so we can estimate that the square root of 107.6 is slightly more than 10.

Use the formula: We can use the following formula to improve our estimate:

x(n+1) = (x(n) + a/x(n))/2

where x(n) is the nth estimate of the square root of a, and a is the number we want to find the square root of.

Apply the formula: Using the estimate of 10, we get:

x(1) = (10 + 107.6/10)/2 = 10.38

Now we use this new estimate to get a better one:

x(2) = (10.38 + 107.6/10.38)/2 = 10.371

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Kara's Custom Tees will spend $540 to produce 8 shirts.

To determine the total expenses encountered by Kara's Custom Tees, we can use the equation:

C(x) = fixed costs + (variable costs per shirt) * x

In this case, the fixed costs are $500 and the variable costs per shirt are $5. Therefore, the equation becomes:

C(x) = 500 + 5x

To calculate the cost of producing 8 shirts, we substitute x = 8 into the equation:

C(8) = 500 + 5 * 8

= 500 + 40

= $540

Therefore, Kara's Custom Tees will spend $540 to produce 8 shirts.

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Step-by-step explanation:

Volume of the square pyramid = 1/3 * base area * height

                                                    = 1/3 (3 x 3) * 4 = 12 yd^3

Answer:

5376

Step-by-step explanation:

volume of prism = 8 X 6 X 14 = 672.

volume of cube = 1/2 X 1/2 X 1/2 = 1/8.

672 / (1/8)

= 672 X (8/1)

= 5376.

5376 cubes will fit into it

Answer:

[tex]2-\sqrt{3}[/tex]

Step-by-step explanation:

Find the exact value of the expression, tan(15°).

The method I am about to show you will allow you to solve this problem without any tables or calculators. Although, memorizing the unit circle and trigonometric identities is required.

[tex]\tan(15 \textdegree)\\\\\Longrightarrow \tan(\frac{30 \textdegree}{2} )\\\\\text{Use the half-angle identity:} \ \tan(\frac{A}{2})=\pm \sqrt{\frac{1-\cos(A)}{1+\cos(A)} }=\frac{\sin(A)}{1+\cos(A)} =\frac{1-\cos(A)}{\sin(A)} \\\\\Longrightarrow\frac{1-\cos(30 \textdegree)}{\sin(30 \textdegree)} \\\\\text{From the unit circle:} \ \cos(30 \textdegree)=\frac{\sqrt{3} }{2} \ \text{and} \ \sin(30 \textdegree)=\frac{1}{2}\\[/tex]

[tex]\Longrightarrow \frac{1-\frac{\sqrt{3} }{2}}{\frac{1}{2}}\\\\\Longrightarrow 2(1-\frac{\sqrt{3} }{2})\\\\\therefore \boxed{\boxed{\tan(15 \textdegree)=2-\sqrt{3} }}[/tex]

Thus, the problem is solved.

Answer:

[tex]\tan 15^{\circ} = 2 - \sqrt{3}[/tex]

Step-by-step explanation:

To find the exact value of tan 15°, we can use trigonometric identities and the unit circle.

We know that tan(x) can be expressed as the ratio of sin(x) and cos(x). We can also write 15° as (60° - 45°).

Therefore, tan 15° can be expressed as:

[tex]\tan15^{\circ}=\tan(60^{\circ}-45^{\circ})=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}[/tex]

Now use the trigonometric angle identities to rewrite the ratio in terms of sin 60°, cos 60°, sin 45° and cos 45°.

[tex]\boxed{\begin{minipage}{6.5 cm}\underline{Trigonometric Angle Identities}\\\\$\sin (A - B)=\sin A \cos B - \cos A \sin B$\\\\$\cos (A - B)=\cos A \cos B + \sin A \sin B$\\\end{minipage}}[/tex]

Therefore:

[tex]\begin{aligned}\tan15^{\circ}&=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}\\\\&=\dfrac{\sin60^{\circ}\cos45^{\circ}-\cos60^{\circ}\sin45^{\circ}}{\cos 60^{\circ} \cos 45^{\circ}+ \sin 60^{\circ}\sin 45^{\circ}}\end{aligned}[/tex]

In the unit circle, the cosine of an angle is represented by the x-coordinate of a point on the circle, and the sine of an angle is represented by the y-coordinate of that same point → (x, y) = (cos θ, sin θ). Therefore, we can use the unit circle to identity the values of sin 60°, cos 60°, sin 45° and cos 45°:

[tex]\sin 60^{\circ}=\dfrac{\sqrt{3}}{2}[/tex]

[tex]\cos 60^{\circ}=\dfrac{1}{2}[/tex]

[tex]\sin 45^{\circ}=\dfrac{\sqrt{2}}{2}[/tex]

[tex]\cos 45^{\circ}=\dfrac{\sqrt{2}}{2}[/tex]

Substitute these into the equation and simplify:

[tex]\begin{aligned}\tan15^{\circ}&=\dfrac{\sin(60^{\circ}-45^{\circ})}{\cos(60^{\circ}-45^{\circ})}\\\\&=\dfrac{\sin60^{\circ}\cos45^{\circ}-\cos60^{\circ}\sin45^{\circ}}{\cos 60^{\circ} \cos 45^{\circ}+ \sin 60^{\circ}\sin 45^{\circ}}\\\\&=\dfrac{\dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}-\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}}{\dfrac{1}{2}\cdot \dfrac{\sqrt{2}}{2}+ \dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{2}}{2}}\\\\\end{aligned}[/tex]

           [tex]\begin{aligned}&=\dfrac{\dfrac{\sqrt{2}}{2} \left(\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}\right)}{\dfrac{\sqrt{2}}{2} \left(\dfrac{1}{2}+ \dfrac{\sqrt{3}}{2}\right)}\\\\&=\dfrac{\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}}{ \dfrac{1}{2}+ \dfrac{\sqrt{3}}{2}}\\\\&=\dfrac{\dfrac{\sqrt{3}-1}{2}}{\dfrac{1+\sqrt{3}}{2}}\\\\&=\dfrac{\sqrt{3}-1}{1+\sqrt{3}}\end{aligned}[/tex]

Simplify further by multiplying the numerator and denominator by the conjugate of the denominator:

           [tex]\begin{aligned}&=\dfrac{\sqrt{3}-1}{1+\sqrt{3}}\cdot \dfrac{1-\sqrt{3}}{1-\sqrt{3}}\\\\&=\dfrac{(\sqrt{3}-1)(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}\\\\&=\dfrac{\sqrt{3}-3-1+\sqrt{3}}{1-\sqrt{3}+\sqrt{3}-3}\\\\&=\dfrac{2\sqrt{3}-4}{-2}\\\\&=-\sqrt{3}+2\\\\&=2-\sqrt{3}\end{aligned}[/tex]

Therefore, the exact value of tan 15° is (2 - √3).

The equation of the line is y = -4x - 11.

The equation of the line is y = 3x + 7.

The equation of the line is y = (1/2)x + 12.

We have,

A)

The equation of the line with gradient m = -4 that passes through the point (-2,-3) is:

y - y1 = m(x - x1)

y - (-3) = -4(x - (-2))

y + 3 = -4(x + 2)

y + 3 = -4x - 8

y = -4x - 11

B)

The equation of the line with gradient m=3 that passes through the point (-2,1) is:

y - y1 = m(x - x1)

y - 1 = 3(x - (-2))

y - 1 = 3(x + 2)

y = 3x + 7

C)

The equation of the line with gradient m=1/2 that passes through the point (-4,10) is:

y - y1 = m(x - x1)

y - 10 = (1/2)(x - (-4))

y - 10 = (1/2)(x + 4)

y = (1/2)x + 12

Therefore,

The equation of the line is y = -4x - 11.

The equation of the line is y = 3x + 7.

The equation of the line is y = (1/2)x + 12.

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The area of the circle is A = 154 m²

Given data ,

Let the diameter of the circle be d = 14 m

So , the radius of the circle is r = d/2

r = 7 m

Now , area of circle is A = πr²

On simplifying , we get

A = ( 3.14 ) ( 7 )²

A = 154 m²

Hence , the area of circle is A = 154 m²

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The location of the angles and the parallel lines [tex]\overline{BE}[/tex] and [tex]\overline{ADE}[/tex] indicates;

(a) ΔABC ~ ΔECA by Angle Angle similarity

(b) The ratio of corresponding sides in similar triangles indicates; BC/CA = AB/EC

Parallel lines are lines that continues indefinitely, maintaining the same distance between each other.

9. The specified dimensions of the geometric figures are;

[tex]\overline{EC}[/tex] is a tangent to the circle O

[tex]\overline{AC}[/tex] is a diameter of the circle

[tex]\overline{BC}[/tex] is parallel to secant [tex]\overline{ADE}[/tex]

Therefore, the angle ∠ABC is a right angle (Angle at the circumference formed by the diameter of a circle

∠ACE = 90° (The tangent is perpendicular to the radius of a circle)

∠ABC ≅ ∠ACE (Definition of congruent angles)

m∠ABC = m∠ACE = 90° (Definition of congruence)

∠BCA ≅ ∠EAC (Alternate interior angles)

(b) The similarity between the triangles and the ratio of the corresponding sides indicates;

BC/CA = AC/AE

Therefore

Segment BC in triangle ΔABC corresponds to segment CA in triangle ΔECA

Segment CA in triangle ΔABC corresponds to segment AE in triangle ΔECA

Which indicates;

Segment AB in triangle ΔABC corresponds to segment EC in triangle ΔECA

The ratio of the corresponding sides is therefore;

BC/CA = AB/EC

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Answer:

Step-by-step explanation:

12

Answer:

get help from a tutor if you need help really fast :)

Step-by-step explanation:

It is a right triangle is the ratio of the length of one leg to the

length of the other leg = x/x =1/1. C.

In a 45-45-90 right triangle, the two acute angles are each 45 degrees, and the right angle is 90 degrees.

The special property of a 45-45-90 triangle is that its two legs are congruent, meaning they have the same length.

Let's denote the length of each leg as "x."

To determine the ratio of the length of one leg to the length of the other leg, we can compare the lengths.

Let's call the length of one leg "a" and the length of the other leg "b."

Since the two legs are congruent, we have a = b = x.

The ratio of the length of one leg to the length of the other leg is a:b = x:x.

Simplifying the ratio, we have a:b = 1:1.

This means that the length of one leg is equal to the length of the other leg in a 45-45-90 right triangle.

To recap:

Both legs have the same length, which is a property unique to this type of triangle.

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The solution to the system of equations are ( 27/13 , -60/13 )

Given data ,

Let the system of equations be A and B

where 2x + ( 1/4 )y = 3   be equation (1)

And , ( 2/3 )x - y = 6   be equation (2)

On simplifying , we get

Multiply by 3 on equation (2) , we get

2x - 3y = 18   be equation (3)

Subtracting equation (1) from (3) , we get

-3y - (1/4)y = 15

-13/4y = 15

Divide by -13/4 on both sides , we get

y = -60/13

Now , the value of x is given by

2x + ( 1/4 ) ( -60/13 ) = 3

2x - ( 60/52 ) = 3

Adding 60/52 on both sides , we get

2x = 3 + 60/52

2x = 216/52

Divide by 2 on both sides , we get

x = 216/104

On simplifying , we get

x = 27/13

Hence , the solution is ( 27/13 , -60/13 )

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Based on the calculations, the yield on this corporate bond is found as 9.14%

Given as For a $1000 face value purchased at a discount price of $850, if it pays 6% fixed interest for the duration of the bond is the yield on a corporate bond mathematically given as

Yield = 6.5%

Interest paid = value of bond x Interest rate

Interest paid = 1000 * 6%

Interest paid = 60

Therefore

Yield = Interest paid / Price paid

Yield = (60 / 850)x 100

Yield = 9.14%

In conclusion, the yield on a corporate bond is

Yield = 9.14%

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Answer: 7.06

Step-by-step explanation: accelus

Step-by-step explanation:

Domain : (-5,4]

Range [-2,0)

The probability that the selection will be P(pink, blue) is: 9/64

The total number of sections on the spinner are 8 sections.

Now, out of the 8 sections, the divisions are as follows:

Yellow sections = 2

Blue sections = 3

Pink sections = 3

Thus:

P(first is pink) = 3/8

P(second is blue) = 3/8

Thus:

P(pink, blue) = (3/8) * (3/8)

= 9/64

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The statement, "You can prove two triangles are similar using an AA similarity theorem" is true while the statement "If the side length of similar figures have a ratio of m/n then the volume will have a ratio of n m³/n³. The correct ratio is (m/n)³" is false.

Visual Match is a term used to describe the degree of similarity or resemblance between two visual elements or objects.

From the given question, we can arrange the following statements as below:

TRUE

- You can prove two triangles are similar using an AA similarity theorem.

- If the side length of similar figures have a ratio of m/n then the surface area will have a ratio of (m/n)².

- If the side length of similar figures have a ratio of m/n then the area will have a ratio of (m/n)².

FALSE

- If the side length of similar figures have a ratio of m/n then the volume will have a ratio of n m³/n³. The correct ratio is (m/n)³.

- If the side length of similar figures have a ratio of 2/3 then the perimeter will have a ratio of 2/3. The correct ratio is 2/3.

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Using the commutative property we can get an equation that depends on the sum of the two variables:

(M + N) = (62 - 2N)/29

The commutative property of the multiplication and sum means that we can permutate the variables:

A*B = B*A

A + B = B + A

In this case, we have the equation:

29*M = 31*(N + 2)

We can distribute the product to get:

29*M = 31*N + 31*2

29*M = 31*N + 62

Now we want to find the value of the sum of the variables, because we have only one equation we can't do that (two variables means that we need two equations) But we can write an expression that depends on the sum of the two variables.

29M + 31N = 62

And now use the commutative property to write:

(29M + 29N) + 2N = 62

(M + N) = (62 - 2N)/29

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The operations between two functions:

Case 1: f(x) + g(x) = 2 - x - x²

Case 2: g(x) - f(x) = x² - x

Case 3: g(x) · f(x) = (1 - x) · (1 - x²) = 1 - x - x² + x³

Case 4: f(x) - g(x) = x - x²

In this problem we need to perform operations between two functions, one operator for each case. There are three operations used in this problem:

Addition

(f + g) (x) = f(x) + g(x)

Subtraction

(f - g) (x) = f(x) - g(x)

Multiplication

(f · g) (x) = f(x) · g(x)

If we know that f(x) = 1 - x² and g(x) = 1 - x, then the operations between functions are:

Case 1

f(x) + g(x) = 2 - x - x²

Case 2

g(x) - f(x) = x² - x

Case 3

g(x) · f(x) = (1 - x) · (1 - x²) = 1 - x - x² + x³

Case 4
f(x) - g(x) = x - x²

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The publisher must produce and sell approximately 4,857 books to break even.

Let's denote the number of books produced and sold as "x."

The total fixed cost is given as $51,000.

The variable cost per book is $1.50, and since x books are produced and sold, the total variable cost would be 1.50x.

The selling price per book is $12, and since x books are sold, the total revenue would be 12x.

To break even, the total revenue must equal the total cost:

Total Revenue = Total Cost

12x = 51,000 + 1.50x

Subtracting 1.50x from both sides:

12x - 1.50x = 51,000

10.50x = 51,000

Dividing both sides by 10.50:

x = 51,000 / 10.50

x ≈ 4,857

Therefore, the publisher must produce and sell approximately 4,857 books to break even.

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The average cost function for the given total cost function is AC(x) = 2,100/x + 4 - 0.0003x.

To find the average cost function C, we need to divide the total cost function C(x) by the quantity of output produced, which is represented by x.
The formula for average cost (AC) is:
AC(x) = C(x) / x
Plugging in the given values, we have:
AC(x) = (2,100 + 4x - 0.0003x2) / x
Simplifying this expression, we get:
AC(x) = 2,100/x + 4 - 0.0003x
Therefore, the average cost function C(x) is:
C(x) = 2,100/x + 4x - 0.0003x2
This function represents the average cost per unit of output, taking into account fixed costs (2,100) and variable costs (4x - 0.0003x) that increase as output increases.
It's important to note that the cost function C(x) is quadratic, which means that the average cost function C(x) will have a U-shaped curve.

This is because initially, as output increases, fixed costs are spread out over a larger quantity of output, leading to a decrease in average cost.

However, at a certain point, the increasing variable costs will outweigh the decreasing fixed costs, causing average cost to increase again.
Overall, knowing the average cost function can be useful for businesses to make decisions about pricing, production levels, and cost management strategies.

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Answer: Nathan's pet cat weighed 6 pounds. Then the cat gained 0.3 pounds. How much does the cat weigh now? Nathan's cat would now weigh 6.3 pounds.

Step-by-step explanation: 6+0.3=6.3

Answer:

6.3

Step-by-step explanation:

It's simple, just do 6 + 0.3, and you get 6.3!

The logarithmic expressions when expanded are

1. [tex]\log2^{8yz}[/tex] = 8xyz log(2)

2. [tex]\log9^{81x/y}[/tex] = 81x/y log(9)

3. [tex]\log5^{x^3}[/tex] = x³ log(5)

4. [tex]\log6^{\sqrt[3]{x}}[/tex] = ∛x log(6)

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

The logarithmic expressions

The logarithmic expressions can be expanded using power rule of logarithm which states that

logaᵇ = b log(a)

Using the above as a guide, we have the following:

[tex]\log2^{8yz}[/tex] = 8xyz log(2)

[tex]\log9^{81x/y}[/tex] = 81x/y log(9)

[tex]\log5^{x^3}[/tex] = x³ log(5)

[tex]\log6^{\sqrt[3]{x}}[/tex] = ∛x log(6)

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The probability that the mean amount of cereal in 5 randomly selected boxes is at most 9.65 ounces is 0.4808.

The Central limit theorem is used to find the probability

Data given:

sample size = 5.

mean, μ = 9.70 ounces

standard deviation, σ = 0.03 ounce.

To calculate the probability, we determine the z-score corresponding to the sample mean of 9.65 ounces using the z-score formula.

x is the sample mean,

μ is the population mean,

σ is the population standard deviation, and

n is the sample size.

For the sample mean of 9.65 ounces in 5 boxes:

z = (9.65 - 9.70) / (0.03 / √5)

z ≈ -0.05 / (0.03 / √5)

Using a calculator, we find that the probability is approximately 0.4801.

Therefore, the probability that the mean amount of cereal in 5 randomly selected boxes is less than 9.65 ounces is approximately 0.4801.

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