Lactation promotes a temporary loss of bone mass to provide adequate amounts of calcium for milk production. A paper gave the following data on total body bone mineral content (TBBMC) (g) for a sample both during lactation (L) and in the postweaning period (P).

Subject
1 2 3 4 5 6 7 8 9 10
L 1928 2549 2825 1924 1628 2175 2114 2621 1843 2541
P 2126 2885 2895 1942 1750 2184 2164 2626 2006 2627

Required:
a. Does data suggest that true average total body bone mineral content during postweaning exceeds that during lactation by more than 25 g? State and test the appropriate hypotheses using a significance level of 0.05. [Note: The appropriate normal probability plot shows some curvature but not enough to cast substantial doubt on a normality assumption.]

b. Calculate a lower confidence bound using a 95% confidence level for the true average difference between TBBMC during postweaning and lactation.

Answer:

B: I think

Step-by-step explanation:

correct me if im wrong

The line PQ is an angle bisector because it divides the angle P into two equal half option (B) angle bisector is correct.

When two lines or rays converge at the same point, the measurement between them is called a "Angle."

We have a triangle shown in the picture.

As we know,  

in terms of geometry, the triangle is a three-sided polygon with three edges and three vertices. The triangle's interior angles add up to 180°.

From the figure the segment PQ divides the angle into two equal half.

From the definition of the angle bisector, the angle bisector can be defined as a line segment that divides the angle into two half.

Angle P = 40 + 40 = 80 degrees

Thus, the line PQ is an angle bisector because it divides the angle P into two equal half option (B) angle bisector is correct.

Learn more about the angle here:

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9514 1404 393

Answer:

  7,6 cm

Step-by-step explanation:

The law of sines can be used to find the length AB.

  AB/sin(C) = BC/sin(A)

A = 180° -48° -52° = 80°

  AB = BC·sin(C)/sin(A) = 12,8·sin(52°)/sin(80°)

The sine function can be used to find AD from AB.

  AD/AB = sin(48°)

  AD = AB·sin(48°) = 12,8·sin(48°)sin(52°)/sin(80°)

  AD ≈ 7,61 cm

__

The dimension of interest is ha in the attachment, the height from vertex A.

============================================================

Explanation:

Let y be the length of the vertical dashed red line in the drawing. More specifically, this dashed line is an altitude.

Along the top, the entire segment is 400 units long. The right piece is 144 units long, so the left piece is 400-144 = 256 units long.

The triangles are similar, allowing us to set up a proportion like so:

144/y = y/256

144*256 = y*y

36864 = y^2

y^2 = 36864

y = sqrt(36864)

y = 192

So this is the length of that vertical dashed red line.

--------------------------------

Now shift your attention solely on the smaller triangle on the right side. It is a right triangle with legs 144 and 192. The hypotenuse is x.

We can use the pythagorean theorem to find x.

a^2 + b^2 = c^2

c = sqrt( a^2 + b^2 )

x = sqrt( 144^2 + 192^2 )

x = 240

240.

Let y be the length of the vertical dashed red line in the drawing. More specifically, this dashed line is an altitude.

Along the top, the entire segment is 400 units long. The right piece is 144 units long, so the left piece is 400-144 = 256 units long.

The triangles are similar, allowing us to set up a proportion like so:

144/y = y/256

144*256 = y*y

36864 = y^2

y^2 = 36864

y = sqrt(36864)

y = 192

So this is the length of that vertical dashed red line.

Now shift your attention solely to the smaller triangle on the right side. It is a right triangle with legs 144 and 192. The hypotenuse is x.

We can use the Pythagorean theorem to find x.

a^2 + b^2 = c^2

c = sqrt( a^2 + b^2 )

x = sqrt( 144^2 + 192^2 )

x = 240

Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a2 + b2 = c2.

Learn more about the Pythagorean theorem at

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Answer:

The answer is "18".

Step-by-step explanation:

Amount of different methods a resident could get an advertisement:

[tex]=2\times 3 \times 3\\\\= 18[/tex]

Answer:

The Answer is 18

Step-by-step explanation:  

2*3*3

Answer:

The rental rate for the Fords is of $12 per day.

Step-by-step explanation:

This question is solved using a system of equations.

I am going to say that:

x is the cost of a Nissan.

y is the cost of a Ford.

z is the cost of a Chevrolet.

Three Nissans, two Fords, and four Chevrolets can be rented for $106 per day.

This means that:

[tex]3x + 2y + 4z = 106[/tex]

Two Nissans, four Fords, and three Chevrolets cost $107 per day

This means that:

[tex]2x + 4y + 3z = 107[/tex]

Four Nissans, three Fords, and two Chevrolets cost $102 per day.

This means that:

[tex]4x + 3y + 2z = 102[/tex]

From the first equation:

[tex]4z = 106 - 3x - 2y[/tex]

[tex]2z = 53 - 1.5x - y[/tex]

[tex]z = 26.5 - 0.75x - 0.5y[/tex]

Replacing into the third equation:

[tex]4x + 3y + 53 - 1.5x - y = 102[/tex]

[tex]2.5x + 2y = 49[/tex]

From the second equation:

[tex]2x + 4y + 3z = 107[/tex]

[tex]2x = 107 - 4y - 3z[/tex]

[tex]x = 53.5 - 2y - 1.5z[/tex]

[tex]x = 53.5 - 2y - 1.5(26.5 - 0.75x - 0.5y)[/tex]

[tex]x - 1.125x = 53.5 - 2y - 39.75 + 0.75y[/tex]

[tex]-0.125x = 13.75 - 1.25y[/tex]

[tex]0.125x = 1.25y - 13.75[/tex]

[tex]x = \frac{1.25y - 13.75}{0.125}[/tex]

[tex]x = 10y - 110[/tex]

Find the rental rate for the Fords.

We have to find y, so:

[tex]2.5x + 2y = 49[/tex]

[tex]2.5(10y - 110) + 2y = 49[/tex]

[tex]25y - 275 + 2y = 49[/tex]

[tex]27y = 324[/tex]

[tex]y = \frac{324}{27}[/tex]

[tex]y = 12[/tex]

The rental rate for the Fords is of $12 per day.

Answer:

Step-by-step explanation:

I honestly have no idea what you mean by answer by formula, but I'm going to give it my best. I began by squaring both sides to get:

(a² - b²) tan²θ = b² and then distributed to get:

a² tan²θ - b² tan²θ = b² and then got the b terms on the side to get:

a² tan²θ = b² + b² tan²θ and then changed the tans to sin/cos to get:

[tex]\frac{a^2sin^2\theta}{cos^2\theta}=b^2+\frac{b^2sin^2\theta}{cos^2\theta}[/tex] and isolated the sin-squared on the left to get:

[tex]a^2sin^2\theta=cos^2\theta(b^2+\frac{b^2sin^2\theta}{cos^2\theta})[/tex] and distributed to get:

***[tex]a^2sin^2\theta=b^2cos^2\theta+b^2sin^2\theta[/tex]*** and factored the right side to get:

[tex]a^2sin^2\theta=b^2(sin^2\theta+cos^2\theta)[/tex] and utilized a trig Pythagorean identity to get:

[tex]a^2sin^2\theta=b^2(1)[/tex] and then solved for sinθ in the following way:

[tex]sin^2\theta=\frac{b^2}{a^2}[/tex] so

[tex]sin\theta=\frac{b}{a}[/tex] This, along with the *** expression above will be important. I'm picking up at the *** to solve for cosθ:

[tex]a^2sin^2\theta=b^2cos^2\theta+b^2sin^2\theta[/tex] and get the cos²θ alone on the right by subtracting to get:

[tex]a^2sin^2\theta-b^2sin^2\theta=b^2cos^2\theta[/tex] and divide by b² to get:

[tex]\frac{a^2sin^2\theta}{b^2}-sin^2\theta=cos^2\theta[/tex] and factor on the left to get:

[tex]sin^2\theta(\frac{a^2}{b^2}-1)=cos^2\theta[/tex] and take the square root of both sides to get:

[tex]\sqrt{sin^2\theta(\frac{a^2}{b^2}-1) }=cos\theta[/tex] and simplify to get:

[tex]\frac{sin\theta}{b}\sqrt{a^2-b^2}=cos\theta[/tex] and go back to the identity we found for sinθ and sub it in to get:

[tex]\frac{\frac{b}{a} }{b}\sqrt{a^2-b^2}=cos\theta[/tex] and simplifying a bit gives us:

[tex]\frac{1}{a}\sqrt{a^2-b^2}=cos\theta[/tex]

That's my spin on things....not sure if it's what you were looking for. If not.....YIKES

Answer:

Step-by-step explanation:

y = ½(x-4)² + 13

y = ½(x² - 8x + 16) + 13

y = ½x² - 4x + 21

Answer:

The 7th term is 10 and the 9th term is 16

(and the common difference d = 3)

Step-by-step explanation:

If by calculate the numbers, you mean the 7th term and 9th term, first, you will determine the common difference.

The nth term of an A.P is given by the formula

Tₙ= a+(n-1)d

Where Tₙ is the nth term

a is the first term

and d is the common difference

From the question,

a = -8

T₇ : T₉ = 5:8

From the formula

T₇ = a + (7-1)d = a + 6d

and T₉ = a + (9-1)d = a + 8d

Then.

a + 6d : a + 8d = 5:8

But a = -8

∴ -8 + 6d : -8 + 8d = 5:8

We can write that

(-8 + 6d) / (-8 + 8d) = 5/8

Cross multiply

8(-8+6d) = 5(-8+8d)

-64 + 48d = -40 + 40d

48d - 40d = -40 + 64

8d = 24

d = 24/8

d = 3

∴ The common difference is 3

Now, for the 7th term

From

T₇ = a + 6d

T₇ = -8 + 6(3)

T₇ = -8 + 18

T₇ = 10

and for the 9th term

T₉ = a + 8d

T₉ = -8 + 8(3)

T₉ = -8 + 24

T₉ = 16

Hence, the 7th term is 10 and the 9th term is 16

Answer: 20.5

Step-by-step explanation:

Cos 43 = X/28

O.73 = x/28

(0.73)(28)=20.5

Answer:

4/5

Step-by-step explanation:

y varies as x

y=kx

k=x/y

Solution :

a). The test is a left tailed test.

b). The sample proportion is :

[tex]$\hat p = \frac{x}{n}$[/tex]

[tex]$\hat p = \frac{17}{258}$[/tex]

  = 0.065

Determining the Z statistics using the formula :

[tex]$Z=\frac{\hat p - p}{\sqrt{\frac{p(1-p)}{n}}}$[/tex]

[tex]$Z=\frac{0.065 - 0.11}{\sqrt{\frac{0.11(1-0.11)}{258}}}$[/tex]

   = -2.31

∴ Z statistics value is -2.31

c). Using the excel function, the P-value is :

P-value = Normsdist(-2.31)

             = 0.0104441

d). The null hypothesis is [tex]$H_0: P = 0.11$[/tex]

    The level of significance is 0.01

    We fail to reject the null hypothesis as the P value is less than or equal to the significant level.

Answer:

PW, BW,  GW, PY,  BY, GY

Step-by-step explanation:

We need to determine the sample space

pink(P), blue (B), and green (G) cards,   (W) and yellow (Y) envelopes

Each color card can match with each color envelope

Start with the white envelopes and each color card

and then the yellow envelopes with each color card

PW   BW  GW

PY  BY   GY

Answer:

0.868 = 86.8% probability that the student passes.

Step-by-step explanation:

Probability of the student passing:

94% of 91%(when the student studies for the test).

14% of 100 - 91 = 9%(when the student does not study for the test). So

[tex]p = 0.94*0.91 + 0.14*0.09 = 0.868[/tex]

0.868 = 86.8% probability that the student passes.

Answer:2

Step-by-step explanation:

9514 1404 393

Explanation:

Definition

The determinant of a square matrix is a single number that is computed (recursively) as the sum of products of the elements of a row or column and the determinants of their cofactors. The determinant of a single element is the value of that element.

The cofactor of an element in an n by n matrix is the (n-1) by (n-1) matrix that results when the row and column of that element are deleted. The "appropriate sign" of the element is applied to the cofactor matrix. The "appropriate sign" of an element is positive if the sum of its row and column numbers is even, negative otherwise. (Rows and columns are considered to be numbered 1 to n in an n by n matrix.)

Uses

The inverse of a square matrix is the transpose of the cofactor matrix, divided by the determinant. Hence if the determinant is zero, the inverse matrix is undefined. This means any system of equations the matrix might represent will have no distinct solution. (There may be zero solutions, or there may be an infinite number of solutions. The determinant by itself cannot tell you which.)

Cramer's Rule for the solution of linear systems of equations specifies that the value of any given variable is the ratio of the determinants of two matrices. The numerator matrix is the original matrix with the coefficients of the variable replaced by the constants in the standard-form equations; the denominator matrix is the original coefficient matrix. This rule lets you solve a system of 3 equations in 3 variables by computing 3+1 = 4 determinants, for example.

Let's look at an example.

If we wanted to solve this system of equations

[tex]\begin{cases}2x-y = 2\\x+y = 7\end{cases}[/tex]

Then it's equivalent to solving this matrix equation

[tex]\begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\7\end{bmatrix}[/tex]

We can then further condense that into the form

[tex]Aw = B[/tex]

Where,

[tex]A = \begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\\\\w = \begin{bmatrix}x\\y\end{bmatrix}\\\\B = \begin{bmatrix}2\\7\end{bmatrix}[/tex]

------------------------------------------

To solve the matrix equation Aw = B, we could compute the inverse matrix [tex]A^{-1}[/tex] and left-multiply both sides by this to isolate w.

So we'd go from [tex]Aw=B[/tex] to [tex]w = A^{-1}*B[/tex]. The order of multiplication is important.

For any 2x2 matrix of the form

[tex]P = \begin{bmatrix}a & b\\c & d\end{bmatrix}[/tex]

its inverse is

[tex]P^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]

Notice the expression ad-bc in the denominator of that fractional term outside. This [tex]ad-bc[/tex] expression represents the determinant of matrix P. Some books may use the notation "det" to mean "determinant"

[tex]P^{-1} = \frac{1}{\det(P)}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]

or you may see it written as

[tex]P^{-1} = \frac{1}{|P|}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}[/tex]

Those aren't absolute value bars, even if they may look like it.

Based on that, we can see that the determinant must be nonzero in order to compute the inverse of the matrix. Consequently, the determinant must be nonzero in order for Aw = B to have one solution.

If the determinant is 0, then we have two possibilities:

So a zero determinant would have to be investigated further as to which outcome would occur.

------------------------------------------

Let's return to the example and compute the inverse (if possible).

[tex]A = \begin{bmatrix}2 & -1\\1 & 1\end{bmatrix}\\\\A^{-1} = \frac{1}{2*1 - (-1)*1}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}\\\\A^{-1} = \frac{1}{3}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}\\\\[/tex]

In this case, the inverse does exist.

This further leads to

[tex]w = A^{-1}*B\\\\w = \frac{1}{3}\begin{bmatrix}1 & 1\\-1 & 2\end{bmatrix}*\begin{bmatrix}2\\7\end{bmatrix}\\\\w = \frac{1}{3}\begin{bmatrix}1*2+1*7\\-1*2+2*7\end{bmatrix}\\\\w = \frac{1}{3}\begin{bmatrix}9\\12\end{bmatrix}\\\\w = \begin{bmatrix}(1/3)*9\\(1/3)*12\end{bmatrix}\\\\w = \begin{bmatrix}3\\4\end{bmatrix}\\\\\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}3\\4\end{bmatrix}\\\\[/tex]

This shows that the solution is (x,y) = (3,4).

As the other person pointed out, you could use Cramer's Rule to solve this system. Cramer's Rule will involve using determinants and you'll be dividing over determinants. So this is another reason why we cannot have a zero determinant.

Step-by-step explanation:

[tex]what \: is \: the \: value \: of \: 5 \times 13 = 65 \: is \: the \: answer[/tex]

Answer:

65

Step-by-step explanation:

[tex] \sqrt{25 \times 169} [/tex]

[tex] \sqrt{5 \times 5 \times 13 \times 13} [/tex]

[tex]5 \times 13[/tex]

[tex]65[/tex]

I hope this helps you

The area bounded by a chord and arc it intercepts is known as a segment of a circle segment of a circle

The area of the circle enclosed by line PL and arc PL is approximately 37.62 square units

The reason the above value is correct is as follows:

The given parameters in the question are;

The radius of the circle, r = 11

The length of the chord PL = 16

The measure of angle ∠PAL = 93°

Required:

The area of part of the circle enclosed by chord PL and arc PL

Solution:

The shaded area of the given circle is the minor segment of the circle enclosed by line PL and arc PL

The area of a segment of a circle is given by the following formula;

Area of segment = Area of the sector - Area of the triangle

Area of segment = Area of minor sector APL - Area of triangle APL

Area of minor sector APL:

Area of a sector = (θ/360)×π·r²

Where;

r = The radius of the circle

θ = The angle of the sector of the circle

Plugging in the the values of r and θ, we get;

Area of the minor sector APL = (93°/360°) × π × 11² ≈ 98.2 square units

Area of Triangle APL:

Area of a triangle = (1/2) × Base length × Height

Therefore;

The area of ΔAPL = (1/2) × 16 × 11 × cos(93°/2) ≈ 60.58 square units

Required shaded area enclosed by line PL and arc PL:

Therefore, the area of shaded segment enclosed by line PL and arc PL is found as follows;

Area of the required segment PL ≈ (98.2 - 60.58) square units = 37.62 square units

The area of the circle enclosed by line PL and arc PL ≈ 37.62 square units

Learn more about the finding the area of a segment can be found here:

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The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.

The calculation of the area between line segment PL and circle arc PL is described below:

1) Calculation of the area of the circle arc.

2) Calculation of the area of the triangle.

3) Subtracting the area found in 2) from the area found in 1).

Step 1:

The area of a circle arc is determined by the following formula:

[tex]A_{ca} = \frac{\alpha\cdot \pi\cdot r^{2}}{360}[/tex] (1)

Where:

[tex]A_{ca}[/tex] - Area of the circle arc.

[tex]\alpha[/tex] - Arc angle, in sexagesimal degrees.

[tex]r[/tex] - Radius.

If we know that [tex]\alpha = 93^{\circ}[/tex] and [tex]r = 11[/tex], then the area of the circle arc is:

[tex]A_{ca} = \frac{93\cdot \pi\cdot 11^{2}}{360}[/tex]

[tex]A_{ca} \approx 98.201[/tex]

Step 2:

The area of the triangle is determined by Heron's formula:

[tex]A_{t} = \sqrt{s\cdot (s-l)\cdot (s-r)^{2}}[/tex] (2)

[tex]s = \frac{l + 2\cdot r}{2}[/tex]

Where:

[tex]A_{t}[/tex] - Area of the triangle.

[tex]r[/tex] - Radius.

[tex]l[/tex] - Length of the line segment PL.

If we know that [tex]l = 16[/tex] and [tex]r = 11[/tex], then the area of the triangle is:

[tex]s = \frac{16+2\cdot (11)}{2}[/tex]

[tex]s = 19[/tex]

[tex]A_{t} = \sqrt{19\cdot (19-16)\cdot (19-11)^{2}}[/tex]

[tex]A_{t} \approx 60.399[/tex]

Step 3:

And the area between the line segment PL and the circle arc PL is:

[tex]A_{s} = A_{ca}-A_{t}[/tex]

[tex]A_{s} = 98.201 - 60.399[/tex]

[tex]A_{s} = 37.802[/tex]

The area of the circle enclosed by line segment PL and circle arc PL is 37.80 square units.

Answer:

18

Step-by-step explanation:

7-5=2

2x9=18

Answer:

Step-by-step explanation:

(-∞,2)

Answer:

a. The average daily amount of time an adult spends on the internet is different than 4.5 hours.

Step-by-step explanation:

An internet provider states that, on average, the daily amount of time an adult spends on the internet is 4.5 hours.

At the null hypothesis, it is claimed that the mean is of 4.5 hours, that is:

[tex]H_0: \mu = 4.5[/tex]

A sociologist studying the behaviors of internet consumers believes the average daily amount an adult spends on the internet is different than the amount stated by the internet provider.

At the alternative hypothesis, the sociologist claim, is that the mean is different of 4.5, that is:

[tex]H_1: \mu \neq 4.5[/tex]

Thus, the correct answer is given by option a.

Answer:

y = -1x + 2

Step-by-step explanation:

Answer:

54% ...

Step-by-step explanation:

this is the answer I guess

The confidence level that produces the widest interval is the one with the highest percentage, which is 54%.

Option B is the correct answer.

A z-score also called a standard score is a measure of how many standard deviations a data point is away from the given mean of a distribution.

It measures the unusual or extreme a particular data point is compared to the rest of the distribution

We have,

The width of a confidence interval is proportional to the critical value of the corresponding confidence level.

The critical value is determined by the standard normal distribution or t-distribution, depending on the sample size and whether the population standard deviation is known.

In general,

The wider the confidence interval, the less precise the estimate of the population means.

Therefore, we want to choose the confidence level that produces the widest interval, which corresponds to the largest critical value.

For a given sample size,

The critical value increases as the confidence level increases.

For example, the critical value for a 95% confidence level is larger than the critical value for a 90% confidence level.

Therefore,

The confidence level that produces the widest interval is the one with the highest percentage, which is 54%.

Learn more about z-score here:

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Answer:

The width is:

[tex]-2+\sqrt{26}\text{ meters}\text{ }(\text{or approximately 3.0990 meters})[/tex]

And the length is:

[tex]2+\sqrt{26}\text{ meters}\text{ } (\text{or approximately 7.0990 meters})[/tex]

Step-by-step explanation:

Recall that the area of a rectangle is given by:

[tex]\displaystyle A = w\ell[/tex]

Where w is the width and l is the length.

We are given that the length of a rectangle is four meters longer than the width. Thus:

[tex]\ell = w + 4[/tex]

And we also know that the area of the rectangle is 22 square meteres.

Substitute:

[tex](22)=w(w+4)[/tex]

Distribute and isolate the equation:

[tex]w^2+4w-22=0[/tex]

The equation isn't factorable, so we can instead use the quadratic formula:

[tex]\displaystyle w = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

In this case, a = 1, b = 4, and c = -22. Substitute:

[tex]\displaystyle w = \frac{-(4)\pm\sqrt{(4)^2-4(1)(-22)}}{2(1)}[/tex]

Evaluate:

[tex]\displaystyle\begin{aligned} w &= \frac{-4\pm\sqrt{104}}{2}\\ \\ &=\frac{-4\pm\sqrt{4\cdot 26}}{2} \\ \\ &=\frac{-4\pm2\sqrt{26}}{2} \\ \\ & = -2\pm \sqrt{26} \end{aligned}[/tex]

Thus, our two solutions are:

[tex]w_1=-2+\sqrt{26}\approx 3.0990\text{ or } w_2=-2-\sqrt{26}\approx-7.0990[/tex]

Since the width cannot be negative, we can ignore the second solution.

Since the length is four meters longer than the width:

[tex]\ell = (-2+\sqrt{26})+4=2+\sqrt{26}\text{ meters}[/tex]

Thus, the dimensions of the rectangle are:

[tex]\displaystyle (2+\sqrt{26}) \text{ meters by } (-2+\sqrt{26})\text{ meters}[/tex]

Or, approximately 3.0990 by 7.0990.

I dont know what you mean by the question but according to me.

If y=x^5

y=x^5+3

Then y+3=x^5+3

Answered by Gauthmath must click thanks and mark brainliest

Answer:

A

Step-by-step explanation:

The be the inverse function the domain {4,5,6,7} becomes the range and the range {14,12,10,8} becomes the domain

14 → 4

12 →5

10 →6

8 →7

9514 1404 393

Answer:

  a) RM2256.09 . . . principal paid by 8th payment

  b) RM1791.10 . . . . interest paid by 12th payment

Step-by-step explanation:

First of all, we need to find the payments.

The payment amount is the amount that makes the future value of the series of payments equal to the future value of the loan at the given interest rate.

The future value of a single amount is ...

  FV = P(1 +r)^n . . . . . where r is the annual rate, and n is the number of years in the future

The future value of a series of payments is ...

  FV = P((1 +r)^n -1)/r . . . . . where n is the number of payments of P earning annual rate r

For payments in a series that does not end at the end of the loan, the future value is the product of that of the series and the effect of the accumulation of interest for the remaining time.

__

The first 4 payments will have a future value at the end of the loan period of ...

  s1 = X((1 +0.05)^4 -1)/0.05×(1 +0.05)^11 = X(1.05^15 -1.05^11)/0.05

  s1 = 7.3717764259X

The next 5 payments will have a future value at the end of the loan period of ...

  s2 = 2X((1 +0.05)^5 -1)/0.05×(1 +0.05)^6 = 2X(1.05^11 -1.05^6)/0.05

  s2 = 14.8097486997X

The last 6 payments will have a future value at the end of the loan period of ...

  s3 = 4X((1 +0.05)^6 -1)/0.05 = 27.20765125X

So, the total future value of the series of payments is ...

  payment value = 7.3717764259X +14.8097486997X +27.20765125X

  = 49.3891763756X

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The future value of the loan amount after 15 years is ...

  loan value = 60,000(1 +0.05)^15 = 124,735.69

In order for these amounts to be the same, we must have ...

  49.3891763756X = 124,735.69

  X = 124,735.69/49.3891763756 = 2,525.57

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At this point, it is convenient to use a spreadsheet to find the interest and principal portions of each of the loan payments. (We find the interest charge to be greater than the payment amount for the first 4 payments. So, the loan balance is increasing during those years.)

In the attached, we have shown the interest on the beginning balance, and the principal that changes the beginning balance to the ending balance after each payment. (That is, the interest portion of the payment is on the row above the payment number.)

The spreadsheet tells us ...

A) the principal repaid in the 8th payment is RM2,256.09

B) the interest paid in the 12th payment is RM1,791.10

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Additional comment

The spreadsheet "goal seek" function could be used to find the payment amount that makes the loan balance zero at the end of the term.

We have used rounding to sen (RM0.01) in the calculation of interest payments. The effect of that is that the "goal seek" solution is a payment value of 2525.56707 instead of the 2525.56734 that we calculated above. The value rounded to RM0.01 is the same in each case: 2525.57.