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A linear function table  is a table having a constant rate of  change between  its datapoints  

A linear equation is is an equation that consists of each variable raised with a maximum of one degree. the linear equation in terms of variables x and y can be written in the form  of  y = mx + c, where m is the slope and x is the independent variable, and y is the dependent variable when the data points of the given  table are potted in   a graph, we get a straight  line for a linear function table, a such function must  have a constant  slope in between its points

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Check the picture below.

\(\cfrac{x}{10}=\cfrac{24.5}{14}\implies x=\cfrac{(10)(24.5)}{14}\implies x=\cfrac{245}{14}\implies x=17.5\)

Answer:

V = d / t

Step-by-step explanation:

La velocidad se define como la relación entre la diatancia cubierta por un cuerpo y el tiempo necesario para cubrir esa distancia.

Si el tiempo empleado se da como (t)

La distancia recorrida se da como (d)

Podemos escribir la velocidad (V) en términos de disance y tiempo como;

V = d / t

Answer:

1). g(x) = \(5^{3x}\)

2). g(x) = \(\frac{1}{3}5^{x}\)

3). g(x) = \(3(5)^{x}\)

4). g(x) = \(5^{\frac{1}{3}x}\)

Step-by-step explanation:

Parent function f(x) = \(a^{x}\) when transformed in the form of  \(g(x)=h(a^{\frac{x}{k} })\)

1). If h > 1, function is vertically stretched.

2). If 0 < h < 1, function is vertically compressed.

3). If k > 1, function is horizontally compressed.

4). If 0 < k < 1, function is horizontally stretched.

Parent function of the given functions in the question is f(x) = \(5^{x}\)

g(x)= \(\frac{1}{3}(5^{x})\), parent function is vertically compressed by a factor of \(\frac{1}{3}\).

g(x) = \(5^{3x}\), parent function 'f' is horizontally stretched by a factor of 3.

g(x) = \(5^{\frac{x}{3}}\), parent function 'f' is horizontally compressed by a factor of \(\frac{1}{3}\).

g(x) = \(3(5^{x})\), parent function 'f' is vertically stretched by a factor of 3.

Answer:

Part A)

\(\displaystyle \frac{dy}{dx}=\frac{2xy}{y-x^2}\)

Part B)

\(y=x-2\)

Part C)

\((0, \sqrt{3})\text{ and } (0, -\sqrt3)\)

Part D)

\(\displaystyle \frac{d^2y}{dx^2}_{(1, -1)}=-\frac{1}{2}\)

Step-by-step explanation:

We have the equation:

\(y^2-2x^2y=3\)

Part A)

We want to find the derivative of the equation. So, dy/dx.

Let’s take the derivative of both sides with respect to x. Therefore:

\(\displaystyle \frac{d}{dx}\Big[y^2-2x^2y\Big]=\frac{d}{dx}[3]\)

Differentiate. We will need to implicitly different on the left. The second term will also require the product rule. Therefore:

\(\displaystyle 2y\frac{dy}{dx}-4xy-2x^2\frac{dy}{dx}=0\)

Rearranging gives:

\(\displaystyle \frac{dy}{dx}\Big(2y-2x^2\Big)=4xy\)

Therefore:

\(\displaystyle \frac{dy}{dx}=\frac{4xy}{2y-2x^2}\)

And, finally, simplifying:

\(\displaystyle \frac{dy}{dx}=\frac{2xy}{y-x^2}\)

Part B)

We want to write the equation for the line tangent to the curve at the point (1, -1).

So, we will require the slope of the tangent line at (1, -1). Substitute these values into our derivative. So:

\(\displaystyle \frac{dy}{dx}_{(1, -1)}=\frac{2(1)(-1)}{(-1)-(1)^2}=1\)

Now, we can use the point-slope form:

\(y-y_1=m(x-x_1)\)

Substitute:

\(y-(-1)=1(x-1)\)

Simplify:

\(y+1=x-1\)

So:

\(y=x-2\)

Part C)

If the line tangent to the curve is horizontal, this means that dy/dx=0. Hence:

\(\displaystyle 0=\frac{2xy}{y-x^2}\)

Multiplying both sides by the denominator gives:

\(0=2xy\)

Assuming y is not 0, we can divide both sides by y. Hence:

\(2x=0\)

Then it follows that:

\(x=0\)

Going back to our original equation, we have:

\(y^2-2x^2y=3\)

Substituting 0 for x yields:

\(y^2=3\)

So:

\(y=\pm\sqrt{3}\)

Therefore, two points where the derivative equals 0 is:

\((0, \sqrt{3})\text{ and } (0, -\sqrt3)\)

However, we still have to test for y. Let’s go back. We have:

\(0=2xy\)

Assuming x is not 0, we can divide both sides by x. So:

\(0=2y\)

Therefore:

\(y=0\)

And going back to our original equation and substituting 0 for y yields:

\((0)^2-2x^2(0)=0\neq3\)

Since this is not true, we can disregard this case.

So, our only points where the derivative equals 0 is at:

\((0, \sqrt{3})\text{ and } (0, -\sqrt3)\)

Part D)

Our first derivative is:

\(\displaystyle \frac{dy}{dx}=\frac{2xy}{y-x^2}\)

Let’s take the derivative of both sides again. Hence:

\(\displaystyle \frac{d^2y}{dx^2}=\frac{d}{dx}\Big[\frac{2xy}{y-x^2}\Big]\)

Utilize the quotient and product rules and differentiate:

\(\displaystyle \frac{d^2y}{dx^2}=\frac{(2y+2x\frac{dy}{dx})(y-x^2)-2xy(\frac{dy}{dx}-2x)}{(y-x^2)^2}\)

Let dy/dx=y’. Therefore:

\(\displaystyle \frac{d^2y}{dx^2}=\frac{(2y+2xy^\prime)(y-x^2)-2xy(y^\prime-2x)}{(y-x^2)^2}\)

For (1, -1), we already know that y’ is 1 at (1, -1). Therefore:

\(\displaystyle \frac{d^2y}{dx^2}_{(1, -1)}=\frac{(2(-1)+2(1)(1))((-1)-(1)^2)-2(1)(-1)((1)-2(1))}{((-1)-(1)^2)^2}\)

Evaluate:

\(\displaystyle \frac{d^2y}{dx^2}_{(1, -1)}=-\frac{1}{2}\)

Since t and s are non-negative, the particles will collide after 3/2 seconds and 5/2 seconds respectively.

a) The two lines intersect if and only if there are real numbers t and s such that r(t)=R(s).

Hence,2+2t=6+st=8-2s10+3t=16-s

Subtracting the first two equations gives 2t-s=4

Substituting this into the third equation gives10+3t=16-2t4t=6t=3/2

Substituting t=3/2 into any of the above three equations gives that the lines intersect at the point:

r(3/2)=R(5/2)=<5, 5, 13>

b) The two particles collide if and only if their paths intersect at some point (t, s).

Thus, the two particles collide if and only if there is a point (t, s) such that r(t) = R(s). We just showed in part a) that such a point exists, namely (3/2, 5/2).

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The fractions are matched to the decimals and attached

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split.

Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.

Decimals are integers that have two components, a whole number component and a fractional component, separated by a decimal point.

Decimals are other forms of writing fractions like in the problem

The fractions are written in decimal form and are matched

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It tells you that Fred has three times the score more than Matt.

This is because Matt has x points, Fred tops Matt by 3x points.

Answer:

Hill 1: F(x)  = -(x + 4)(x + 3)(x + 1)(x - 1)(x - 3)(x - 4)

Hill 2: F(x)  = -(x + 4)(x + 3)(x + 1)(x - 1)(x - 3)(x - 4)

Hill 3: F(x) = 4(x - 2)(x + 5)

Step-by-step explanation:

Hill 1

You must go up and down to make a peak, so your function must cross the x-axis six times. You need six zeros.

Also, the end behaviour must have F(x) ⟶ -∞ as x ⟶ -∞ and F(x) ⟶ -∞ as x⟶ ∞. You need a negative sign in front of the binomials.

One possibility is

F(x)  = -(x + 4)(x + 3)(x + 1)(x - 1)(x - 3)(x - 4)

Hill 2

Multiplying the polynomial by -½ makes the slopes shallower. You must multiply by -2 to make them steeper. Of course, flipping the hills converts them into valleys.

Adding 3 to a function shifts it up three units. To shift it three units to the right, you must subtract 3 from each value of x.

The transformed function should be

F(x)  = -2(x +1)(x)(x -2)(x -3)(x - 6)(x - 7)

Hill 3

To make a shallow parabola, you must divide it by a number. The factor should be ¼, not 4.

The zeroes of your picture run from -4 to +7.

One of the zeros of your parabola is +5 (2 less than 7).

Rather than put the other zero at ½, I would put it at (2 more than -4) to make the parabola cover the picture more evenly.

The function could be

F(x) = ¼(x - 2)(x + 5).

In the image below, Hill 1 is red, Hill 2 is blue, and Hill 3 is the shallow black parabola.

The z-scores will remain the same regardless of whether you use inches or centimetres for the height measurements.

The z-scores will not change if you convert the height measurements from inches to centimetres or vice versa. The z-score is a standard score representing the number of standard deviations, a value above or below the mean of a normal distribution.

The z-score is calculated using the formula z = (x - mean)/standard deviation, where x is the value being compared to the mean and standard deviation of the distribution.

Converting the height from inches to centimetres or vice versa will only change the units of measurement, but the relative position of a value within the distribution will remain unchanged.

Therefore, the z-scores will remain the same regardless of whether you use inches or centimetres for the height measurements.

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The value of (1+tan2A)(1-sec A)(1+sec A) is 1.

We can simplify the given expression using the trigonometric identities:

tan2A = 2tanA/(1-tan²A) and sec A = 1/cos A.

To simplify the given expression (1+tan²A)(1-secA)(1+secA), we can start by using trigonometric identities to express the terms in the expression in terms of a single trigonometric function.

Substituting these values, we get:

(1+tan2A)(1-sec A)(1+sec A)

= [1+2tanA/(1-tan²A)][1-1/cos A][1+1/cos A]

= [1-tan²A+2tanA][cos A-1/cos²A]

= [sec²A+2tanA][cos²A-1/cos²A]

= [sec²A+2tanA][sin²A/cos²A]

= [1/cos²A+2sinA/cosA][sin²A/cos²A]

= sin²A/cos²A

= 1

Therefore, the value of (1+tan2A)(1-sec A)(1+sec A) is 1.

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The dissociation curve is a graphical representation of the relationship between the fractional saturation of hemoglobin (Y-axis) and the partial pressure of oxygen (X-axis) under specific conditions. It provides important information about the binding and release of oxygen by hemoglobin.

The dissociation curve for hemoglobin exhibits a sigmoidal (S-shaped) shape. At low partial pressures of oxygen, such as in tissues, hemoglobin has a low affinity for oxygen and only binds a small amount. As the partial pressure of oxygen increases, hemoglobin's affinity for oxygen increases, resulting in a rapid increase in the binding of oxygen molecules. However, once the hemoglobin becomes nearly saturated with oxygen, the curve levels off, indicating that further increases in partial pressure have minimal effects on oxygen binding.

To calculate the fractional saturation of hemoglobin at a given partial pressure of oxygen, you can use the Hill equation:

Y = [O2]^n / ([O2]^n + P50^n)

Where:

Y is the fractional saturation of hemoglobin,

[O2] is the partial pressure of oxygen,

P50 is the partial pressure of oxygen at which hemoglobin is 50% saturated,

n is the Hill coefficient, which represents the cooperativity of oxygen binding.

To determine the P50 value experimentally, the partial pressure of oxygen at which hemoglobin is 50% saturated, you can plot the dissociation curve and identify the point where the curve reaches 50% saturation.

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The absolute value function for this problem is given as follows:

g(x) = |x - 1| - 2.

An absolute value function of vertex (h,k) is defined as follows:

y = a|x - h| + k.

In which a is the leading coefficient.

The coordinates of the vertex for this problem are given as follows:

(1, -2).

As the slope of the line is of 1, the leading coefficient is given as follows:

a = 1.

Hence the absolute value function for this problem is given as follows:

g(x) = |x - 1| - 2.

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Magnitude refers to the severity of the harm, while probability is the likelihood of it occurring. In the given scenario, even though the probability of identifying an individual subject is low, the high magnitude of harm due to the sensitive nature of the information makes it necessary to consider both elements when determining overall risk.

When determining risk, it is important to take into account both the magnitude and probability of harm. The magnitude refers to the severity or level of harm that could occur, while probability refers to the likelihood of harm occurring. In mathematics, the size or size of a mathematical object is a property that determines whether that object is larger or smaller than other objects of its kind. More formally, the size of an object is the result of identifying (or ranking) the category of objects to which it belongs.

In the given scenario, even though the probability of an individual being identified is low, the magnitude of harm is high due to the sensitivity of the information. Therefore, it is crucial to consider both elements when assessing risk in order to make informed decisions and take necessary precautions.

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Answer:

yn +y -xn

Step-by-step explanation:

Mean= sum of numbers ÷number of numbers

x= sum of n numbers ÷n

Multiply both sides by n:

xn= sum of n numbers

sum of n numbers= xn -----(1)

Let the number added be a.

After a has been added,

y= (sum of n numbers +a) ÷(n+1)

Multiply both sides by (n +1):

y(n +1)= sum of n numbers +a

sum of n numbers= y(n +1) -a -----(2)

Substitute (1) into (2):

xn= y(n +1) -a

Expand:

xn= yn +y -a

a= yn +y -xn

Let's check!

Let the numbers be 1, 2, 3 and 4.

mean, x= 10 ÷4= 2.5

n= 4

Let the new number added, a, be 7.

a= 7

mean, y= 17 ÷5= 3.4

a= yn +y -xn

a= 3.4(4) +3.4 -2.5(4)

a= 7

Thus, the expression is true.

Answer:

length = 15

Width = 9

Step-by-step explanation:

In a rectangle opposite sides are equal.

5x - y- 8 =3x +5y + 4

5x - 3x  -y - 5y = 4 + 8

2x - 6y = 12

Divide the entire equation by 2

x  - 3y = 6  --------------(I)

3x + y - 4 = 2x - 6y - 3

3x - 2x + y + 6y = -3 + 4

x + 7y = 1 -------------(II)

Multiply equation (II) by (-1)

(I)            x - 3y = 6

(II)*(-1)    -x - 7y = -1   {Now add. 'x' will be eliminated}

                  -10y= 5

                      y = 5/(-10)

                     y = -0.5

Plugin y = -0.5 in equation (I)

x - 3*(-0.5) = 6

x  + 1.5 = 6

        x = 6 - 1.5

        x = 4.5

Length = 5x - y - 8

           = 5*4.5 - (-0.5) - 8

           = 22.5 + 0.5 - 8

            = 23 - 8

Length = 15

Width = 2x - 6y - 3

         = 2*4.5 - 6*(-0.5) - 3

         =  9 + 3 - 3

Width = 9

The radius and its uncertainty of circle from the following values of L and h.  L = (125.0 ± 5.0) cm, h = (0.51 ± 0.22) cm and  L = (125.0 ± 5.0) cm, h = (57.4 ± 1.2) cm are (12232.9 ± 209.4) cm and (143.1 ± 1.3) cm respectively.

We can use the given equation to calculate the radius of the circle using the values of L and h:

R = L^2 / 2h + h / 2

Substituting the given values with uncertainties, we get:

R = (125.0 cm ± 5.0 cm)^2 / (2 × 0.51 cm ± 0.22 cm) + (0.51 cm ± 0.22 cm) / 2

R = 15158.4 cm^2 / 1.24 cm ± 6.16 cm^2 / cm

R = (12232.9 ± 209.4) cm

Therefore, the radius of the circle is approximately 12232.9 cm with an uncertainty of ±209.4 cm.

We can see that the second term of the equation (h/2) contributes to the final value of the radius. Therefore, it was necessary to use this term in the calculation of R.

Using the same equation with the given values of L and h, we get:

R = L^2 / 2h + h / 2

R = (125.0 cm ± 5.0 cm)^2 / (2 × 57.4 cm ± 1.2 cm) + (57.4 cm ± 1.2 cm) / 2

R = 15158.4 cm^2 / 114.2 cm ± 1.16 cm^2 / cm

R = (143.1 ± 1.3) cm

Therefore, the radius of the circle is approximately 143.1 cm with an uncertainty of ±1.3 cm.

In this case, the second term of the equation (h/2) is much smaller than the first term (L^2/2h) and contributes only a small amount to the final value of the radius. Therefore, in this case, it was not strictly necessary to include the second term in the calculation of R. However, including it can still help to more accurately represent the uncertainty in the final result.

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Answer:

length = 75 ft and width = 25 ft

Step-by-step explanation:

Let W = width of rectangle

Let L = length of rectangle

Area of a rectangle = L x W

Perimeter of a rectangle = 2L + 2W

Therefore, if the perimeter is 200 ft then:

2L + 2W = 200

2(L + W) = 200

    L + W = 100

           L = 100 - W

If the Area is 1875 ft² then:                             1875 = L x W

Substitute the equation for L found above:  1875 = (100 - W) x W

Expand the brackets:                                    1875 = 100W - W²

Subtract 1875 from both sides:  100W - W² - 1875 = 0

Swap signs:                                W² - 100W + 1875 = 0

Factorize:                                       (W - 25)(W - 75) = 0

Therefore, W = 25 or 75

If W = 25, then L = 100 - 25 = 75

If W = 75, then L = 100 - 75 = 25

Therefore, as length > width, length = 75 ft and width = 25 ft

The order of\($gN$\) divides \($k$\), as required.

Let \($g\in G$\) be an arbitrary element and let \($k$\) be the order of \($g$\), that is, \($g^k=e_G$\), the identity element of \($G$\). We want to show that the order of \($gN$\) in \($G/N$\) divides \($k$\).

Consider the coset \($g^iN\in G/N$\) for some positive integer \($i$\). We want to find the smallest positive integer \($j$\) such that \($(gN)^j = g^jN = g^iN$\). Since \($g^iN = g^k(g^i)^kN = g^kN$\), we have \($(gN)^j = g^{jk}N = g^iN$\), which implies \($g^{jk-i}\in N$\).

Since \($N$\) is a normal subgroup of\($G$\), we have \($g^{jk-i}\in N$\) if and only if \($g^{jk-i}g^j=g^{jk}\in N$\). This shows that \($g^{jk}$\) is in the kernel of the canonical homomorphism \($\pi:G\to G/N$\), so\($k$\) is a multiple of the order of\($gN$\) in \($G/N$\).

Therefore, the order of \($gN$\) divides \($k$\), as required.

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Answer:

The answer is 9/x-1

Step-by-step explanation:

Because both terms have the same denominator, then we can subtract the numerators upon the same denominator:

⇒ x+1/x-1 - x-8/x-1 = ((x + 1) - (x - 8))/(x - 1)

                             = (x + 1 - x + 8)/(x - 1)     [expand the brackets and simplify]

                             = 9/(x - 1)

Answer:

y = 6(x + 8)² + 4

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

Using the method of completing the square.

y = 6x² + 96x + 388

The coefficient of the x² term must be 1, so

factor out 6 from 6x² + 96x

y = 6(x² + 16x ) + 388

To complete the square

add/subtract (half the coefficient of the x- term)² to x² + 16x

y = 6(x² + 2(8)x + 64 - 64) + 388

  = 6(x + 8)² - 384 + 388

y = 6(x + 8)² + 4 ← in vertex form

Answer:

The numbers are 36 and 68.

Step-by-step explanation:

Let the smaller number be x.

"The larger number is four less than two times the smaller

number."

The larger number is

2x - 4

The sum of the numbers is x + 2x - 4, or 3x - 4.

"The sum of the numbers is 104."

3x - 4 = 104

3x = 108

x = 36

The smaller number is 36.

The larger number is 2x - 4, or

2x - 4 = 2(36) - 4 = 72 - 4 = 68

The larger number is 68.

Answer: The numbers are 36 and 68.

Answer:

5

Step-by-step explanatio

To measure the volume of an irregular-shaped object, you can use the water displacement method, geometric approximation, or advanced techniques like 3D scanning technology.

To measure the volume of an irregular-shaped object, you can use one of several methods depending on the object's characteristics. Here are a few common approaches:

1. Water Displacement Method:

  - Fill a container with water and measure the initial volume (V1) of the water.

  - Carefully place the irregular object into the container, ensuring it is fully submerged.

  - The object will displace a volume of water equal to its own volume.

  - Measure the final volume (V2) of the water and subtract the initial volume (V1) to determine the volume of the object.

  Volume of the object = V2 - V1

2. Graduated Cylinder Method:

  - If the irregular object is small enough, you can use a graduated cylinder filled with a liquid (such as water) instead of a large container.

  - Carefully lower the object into the graduated cylinder, ensuring it is fully submerged.

  - Read the initial volume of the liquid in the graduated cylinder.

  - Measure the final volume of the liquid after adding the object.

  - The difference between the initial and final volume indicates the volume of the object.

3. Geometric Approximation:

  - If the irregular object has a shape that closely resembles a known geometric shape (e.g., a cone, cylinder, or box), you can approximate its volume using the appropriate formula for that shape.

  - Measure the relevant dimensions of the object, such as height, base radius, or length, depending on the shape.

  - Calculate the volume using the appropriate formula for the geometric shape.

4. 3D Scanning Technology:

  - Advanced techniques involve using 3D scanning technology to create a digital model of the irregular object.

  - Specialized software can then calculate the volume based on the digital model.

It's important to note that the accuracy of these methods depends on the precision of the measurements and any assumptions made in the process. Select the method that suits your object and available resources best, considering factors such as size, material, and the desired level of accuracy.

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If two years ago one egg cost one naira less than last year,  this year an egg cost 2 naira more than last year the cost of the 11 eggs, 2 years ago was the same as the cost of an egg last year $9.

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

Cost of egg last year  = x

Two years ago an egg cost  1 less than last year.

Hence cost of egg  Two years ago = x - 1

the cost of 11 eggs two year ago = 11 (x - 1)

his year an egg costs  2 more than last year .

Hence cost of egg this year = x + 2

cost of eight eggs this year = 8(x + 2

Equate

11 (x - 1) = 8(x + 2)

11x - 11 = 8x + 16

3x  = 27

x = 9

Hence, the cost of the egg last year is $9.

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Answer:

I'm an island boii ️️

Based on the given information, we are told that Harry has $480 and Lee has $960. We are also given the assumption that Joy has the same number of dollars more than Harry as she has less than Lee.

To determine the amount of money Joy has, we can calculate the difference between the amount of money Lee has and the amount of money Harry has.

The amount of money Joy has = Amount of money Lee has - Amount of money Harry has.

Substituting the given values into the equation, we have:

Amount of money Joy has = $960 - $480 = $480.

Therefore, Joy has $480.

The explanation concludes by stating that option B is the correct answer.

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Answer:

(9,10)

Step-by-step explanation:

It went up 6 and over 6, so we need to do that again.

(9, 10)

The probability that X, which follows a normal distribution with a mean (u) of 15.2 and a standard deviation (a) of 0.9, falls between 14.3 and 16.1 is approximately 0.6826, or 68.26%.

In a normal distribution, the area under the curve represents probabilities. To find the probability that X falls between 14.3 and 16.1, we need to calculate the area under the curve between these two values.

First, we convert the values to z-scores by subtracting the mean and dividing by the standard deviation. For 14.3, the z-score is (14.3 - 15.2) / 0.9 = -1. The z-score for 16.1 is (16.1 - 15.2) / 0.9 = 1.

Using a standard normal distribution table or a calculator, we can find that the cumulative probability for a z-score of -1 is 0.1587, and the cumulative probability for a z-score of 1 is 0.8413.

To find the probability between these two z-scores, we subtract the cumulative probability for the lower z-score from the cumulative probability for the higher z-score: 0.8413 - 0.1587 = 0.6826.

Therefore, the probability that X falls between 14.3 and 16.1 is approximately 0.6826, or 68.26%.

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The complete question is :

Assume that X has a normal distribution. The mean is u= 15.2 and the standard deviation is a=0.9. Find the probability that X is between 14.3 and 16.1