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Show that the triangle with vertices A, B, and C is a right triangle.

Answer:

To find (g of f)(x), we need to first find f(x) and then substitute it into g(x).

f(x) = x - 4

Now we can substitute f(x) into g(x):

g(f(x)) = 9f(x) + 4

g(f(x)) = 9(x - 4) + 4

g(f(x)) = 9x - 36 + 4

g(f(x)) = 9x - 32

Therefore, the answer is (A) 9x+32.

Answer:

the answer is d- (x-5)

The length of the lake is equivalent to 295 meters.

We can write that -

tan(32°) = 250/{x}

{x} = 250/tan(32°)

also

tan(68°) = 250/{y}

{y} = 250/tan(68°)

We can write the length of the lake as -

L = x - y

L = 250/tan(32°) -  250/tan(68°)

L = 250{1/tan(32°) - 1/tan(68°)}

L = 250{1/(0.63) - 1/(2.48)}

L = 250{1.58 - 0.40}

L = 250 x 1.18

L = 295 meters

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The solution is, the coordinates of midpoint of AB are (-3/2, -1)

We will use the formula of midpoint ;

(Xm , Ym) = (x₁ + x₂ /2 , y₁ + y₂ /2)

This implies that;

Xm = x₁ + x₂ /2

From the question, x₁ = -4 y₁= 2 and,  x₂ =1, y₂ = -4

substituting the values into;

Xm = x₁ + x₂ /2 and then solve for x_m

x_m = -4 + 1 / 2

x_m = -3/2

Similarly, substituting the values into;

Ym = y₁ + y₂ /2 and then solve for y_m

so. we get,

y_m = 2+ (-4) / 2

or, y_m = -1

Hence the coordinates of midpoint of AB are (-3/2, -1)

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Answer:

Marty has 768 pints of orange juice.

Step-by-step explanation:

To convert gallons to pints, we need to remember that 1 gallon is equal to 128 fluid ounces, and 1 fluid ounce is equal to 1/8 pint.

Given that Marty has 6 gallons of orange juice, we can calculate the total amount in pints as follows:

6 gallons x 128 fluid ounces/gallon x 1/8 pint/fluid ounce = 768 pints

So, Marty has 768 pints of orange juice.

The area of the new shape is 74 squared cm

The area of the new shape is solved by dividing the shape into three sections and then sum up all the areas

The area of the sections are as follows

Section 1: length x width

= 8 x 5

= 40 squared cm

Section 2: length x width

= 5 x 2

= 10 squared cm

Section 3: length x width

= 8 x 3

= 24 squared cm

Total area

= 24 + 10 + 40

= 74 squared cm

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The area of the new shape is 74cm²

The area is the region bounded by the shape of an object. The space covered by the figure or any two-dimensional geometric shape, in a plane, is the area of the shape.

The shape is a combination of two rectangles. One of the rectangles cut out.

The area of the cut out is calculated as;

A = l × w

A = 3 × 2

A = 6cm²

The area of the cut out rectangle = area of the whole rectangle - area of cut out rectangle

= 8×5 -6

= 40-6

= 34cm²

The area of the second rectangle = l×w

= 8× 5

= 40cm²

Therefore area of the new shape = 40+34

= 74cm²

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The simplest radical form of the sine of y/2 is:

sin(y/2) = ±√(2/5)

Here we know that:

cos(y) = 1/5.

We want to find  sin(y/2), so we can use the identity:

sin(y/2) = ±√( ( 1 - cos(y))/2)

Replacing cos(y) there we will get.

sin(y/2) = ±√( ( 1 - 1/5)/2)

sin(y/2) = ±√( ( 4/5/2)

sin(y/2) = ±√(2/5)

That is the simplest radical form.

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"If you share 7 apples equally with 3 people, then there are 2 1/3 apples for each person."The whole number 2 represents the number of apples that each person will get when sharing 7 apples equally with 3 people.

The numerator 1 represents the remaining part of an apple that is left after sharing equally among 3 people. The denominator 3 represents the number of persons with whom the apples are being shared equally.

A possible real-life application for the expression 1 ÷ 1/4 is dividing a pizza into quarters and then sharing one slice of pizza equally among a group of people.

To simplify the expression for sharing, we need to remember that dividing by a fraction is the same as multiplying by its reciprocal. So, 1 ÷ 1/4 is equivalent to 1 × 4/1, which simplifies to 4.

Thus, 1 ÷ 1/4 is equal to 4.

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The probability 0.7977 that the sample mean weight of 12 adult male passengers is greater than 162 lb.

There is a 79.77% chance that the elevator is overloaded

We can use the Central Limit Theorem here, which states that the sample mean of a sufficiently large sample size (n≥30) will be approximately normally distributed with mean μ and standard deviation σ/√n, where μ is the population mean and σ is the population standard deviation.

In this case, n=12, μ=169 lb, and σ=29 lb. Therefore, the sample mean weight of 12 adult male passengers will be approximately normally distributed with mean 169 lb and standard deviation 29/√12 ≈ 8.38 lb.

Now we can standardize this distribution to find the probability that the sample mean weight is greater than 162 lb:

z = (162 - 169) / 8.38 ≈ -0.83

We determine that the probability of z being less than or equal to -0.83 is approximately 0.2023, which can be calculated using a standard normal distribution table or calculator.

Therefore, there is a probability of 1 - 0.2023 ≈ 0.7977 that the sample mean weight of 12 adult male passengers is greater than 162 lb.

This indicates that if the elevator is filled with 12 adult male passengers, there is a 79.77 percent chance that it will be overloaded. The elevator does not appear to be safe for this many passengers, and this is a very high probability

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The calculated time of is that the baseball game was 1 7/12 hour longer than the movie

The given parametes in the question are

Movie = 1 3/4 hours

Baseball = 3 1/3 hours

To calculate how much longer was the baseball game than the movie, we subtract

So, we have

Longer = 3 1/3 - 1 3/4

Evaluate

Longer = 19/12

This gives

Longer = 1 7/12

Hence, the baseball game was 1 7/12 hour longer than the movie

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Answer:

There are 48 different combinations possible with 8 shirts, 2 pairs of jeans, and 3 pairs of shoes.

Step-by-step explanation:

To find the number of different combinations possible with 8 shirts, 2 pairs of jeans, and 3 pairs of shoes, we need to multiply the number of options for each category.

For shirts, there are 8 options.

For jeans, there are 2 options (either wear a pair or not wear a pair).

For shoes, there are 3 options.

To find the total number of combinations, we multiply these options:

8 * 2 * 3 = 48

Therefore, there are 48 different combinations possible with 8 shirts, 2 pairs of jeans, and 3 pairs of shoes.

Answer: 48

Step-by-step explanation:

Total outcomes, you multiply all possibilities.

8 shirts

2 jeans

3 shoes

8*2*3 =48

The area of the given figure is 192.5 ft².

Given is a figure we need to find its area,

By splitting the shape in 2, we can find the area.

Starting with the parallelogram (on the left), the area formula is A = b × h.

b (base) = 11 ft.

h (height) = 7 ft.

A (area) = (11 ft.)(7 ft.)

A = 77 ft.²

Then, the trapezoid's (on the right) area formula is A = [(a + b) ÷ 2] × h.

a (first base) = 8 ft.

b (second base) = (20 - 7) = 13 ft.

h (height) = 11 ft.

A = [(8 ft. + 13 ft.) ÷ 2] × 11 ft.

A = [(21 ft.) ÷ 2] × 11 ft.

A = (10.5 ft.) × 11 ft.

A = 115.5 ft.²

we can add them together to find the total area:

A = 77 ft.² + 115.5 ft.²

A = 192.5 ft.²

Hence, the area of the given figure is 192.5 ft².

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The area of the shape is 120 if the perimeter of the given shape is 44 feet.

The shape is given below.

It is in the shape of a rectangle.

Length = 6x and width = 4x + 2

Perimeter of a rectangle = 2 × (Length + Width)

So,

2 (6x + 4x + 2) = 44

2 (10x + 2) = 44

10x + 2 = 22

10x = 20

x = 2

So, length = 6x = 12

Width = 4x + 2 = 10

Area of the rectangle = Length × Width

                                    = 12 × 10

                                    = 120

Hence the area of the shape is 120.

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Answer:

The value of y in the list 1/2,x,y,3/4 is 2/3. This is because the differences between successive numbers in the list are all the same, and the difference between 1/2 and 3/4 is 2/3.

The solution of "system-of-equations" shown in the graph is (d) There is one unique solution (-3, 1).

In the graph shown, we see that the two lines representing the two equations, intersect at a single point,

So, we can say that the given system-of-equations has a "unique-solution",

On carefully observing the point of intersection, we see that, the "x-coordinate" of intersecting point is "-3", and the "y-coordinate" of the intersecting point is "1",

So, the unique solution of the "system-of-equations" occur at the point (-3,1).

Therefore, the correct option is (d).

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The correct statement is :- The measure of an arc corresponds to the measure of a central angle and an arc length is a fraction of the​ circle's circumference. [option A]

The measure of an arc is calculated according to how much part of the circle is intercepted by the central angle which is corresponding to that arc.

Whereas,

Length of an arc is the measure of a certain part of the circumference of the circle.

Hence the measure of an arc corresponds to the measure of a central angle and an arc length is a fraction of the​ circle's circumference.

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The parabola has vertex of (-6, 11) and a p - value of 4.5.

We can calculate the midpoint between the focus and the directrix point that is directly above or below the focus in order to locate the vertex. This point is at (-6, 11), which is the same as the focus' x-coordinate and the directrix's y-coordinate.

We can use either the distance between the vertex and the directrix or the focus to determine the p-value:

p = ( k - ( - 7  )) / 2

= (2 - (-7) ) / 2

= 9/2

= 4. 5

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The median of the boxplot is 420.

In a boxplot, the median is the line in the Centre of the box.

Given that;

The upper horizontal line of the box is at 490

The Lower horizontal line of the box is at 350

Therefore the median of boxplot

= (upper point + lower point)/2

= (490 + 350)/2

= 420

Hence middle point be at 420.

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The value of the height of the cone is 38 units

The formula for calculating the circumference of a cone is represented as;

Circumference = 2πr

Now, substitute the value to determine the radius, we have;

24π = 2πr

Divide by the coefficient of r

r = 24π/2π

r = 12 units

The formula for the volume of a cone is;

Volume = πr²h/3

Substitute the values

1824 π = π × 144 ×h/3

Multiply the values

1824π = 144πh/3

Make h the subject

h = 38 units

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Answer:

To factor the quadratic equation y = x^2 - 8x - 9, we need to find two numbers that multiply to give -9 and add to give -8 (the coefficient of the x term).

We can start by looking for two factors of -9 that add up to -8. The only pair of factors that works are -9 and 1. This is because (-9) + 1 = -8, and (-9) times 1 is -9.

Now we can use these factors to express the middle term of the quadratic in terms of two separate terms that can be factored by grouping. We can do this by splitting the -8x term into -9x + x:

y = x^2 - 9x + x - 9

Now we can group the first two terms together and the last two terms together:

y = (x^2 - 9x) + (x - 9)

We can factor out x from the first group, and factor out -9 from the second group:

y = x(x - 9) + (-9)(x - 9)

Finally, we can factor out the common factor of (x - 9):

y = (x - 9)(x - 1)

Therefore, the factor form of the quadratic equation y = x^2 - 8x - 9 is (x - 9)(x - 1).

The length of chord LM is determined as 24 inches.

option A.

The length of the chord LM is calculated by applying the following formula as shown below;

Based on intersecting chord theorem, when two chord intersect in a circle, the product of two segment of one chord is equal to the product of two segment of the second chord.

First we need to find the value of x;

3x(x) = 12 (9)

3x² = 108

x² = 108/3

x² = 36

x = √ 36

x = 6

The length of chord LM is calculated as follows;

LM = x + 3x

LM = 6 + 3(6)

LM = 24

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The values of the variables x, y and a in the circles are calculated below

Circle 1

The measure of the angle x is calculated as

28 = 1/2(x - 61)

So, we have

x - 61 = 56

This gives

x = 117

For y, we have

y² = 4 * 18

So, we have

y = 6√2

Circle 2

The measure of the angle x is calculated as

x + 48 = 180

So, we have

x = 132

The length y is calculated as

tan(24) = 7/y

So, we have

y = 7/tan(24)

Evaluate

y = 15.7

Circle 3

This is calculated as

9 * (9 + a) = 8 * 21

So, we have

81 + 9a = 168

So, we have

a = 9.67

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Answer:

The formula for calculating the future value of a present amount with compound interest is:

FV = PV * (1 + r)^n

where FV is the future value, PV is the present value, r is the annual interest rate (as a decimal), and n is the number of compounding periods.

In this case, Nkaiseng wants to invest some money now to retire with R500 300 in 40 years (from 25 to 65). The annual interest rate is 11.5%, which is equivalent to a monthly interest rate of 0.115/12 = 0.00958.

So we have:

FV = PV * (1 + 0.00958)^ (40*12)

500300 = PV * (1.00958)^480

Dividing both sides by (1.00958)^480, we get:

PV = 500300 / (1.00958)^480

PV = 500300 / 20.4524

PV = 24406.53

Therefore, Nkaiseng should invest R24406.53 (rounded up to the nearest rand) now to have R500 300 when she retires at 65 years with an interest rate of 11.5% per annum.

Answer:

Let's assume that the number of children who used the public swimming pool is 'x' and the number of adults who used the pool is 'y'.

From the given information, we can form two equations:

x + y = 454 (equation 1) --> the total number of people who used the pool

1.5x + 2.5y = 1043 (equation 2) --> the total revenue collected from admission fees

We can use these equations to solve for 'x' and 'y'. Let's start by multiplying equation 1 by 1.5 to eliminate 'x':

1.5x + 1.5y = 681 (equation 1 multiplied by 1.5)

1.5x + 2.5y = 1043 (equation 2)

Now, we can subtract equation 1 from equation 2 to eliminate 'x' and solve for 'y':

2.5y - 1.5y = 1043 - 681

y = 224

So, there were 224 adults who used the pool. We can substitute this value into equation 1 to solve for 'x':

x + 224 = 454

x = 230

Therefore, there were 230 children who used the pool.

Answer:

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

The diagonals of a square bisect each other.

It makes all half-diagonals of equal length. Therefore:

Substitute and solve for x:

Hence the value of x is 4.

The line of reflection for the graph is Reflection across x = 6.

Given a graph of the polygon ABCDE.

Vertices of the polygon are :

A(-1, -4), B(-1, -1), C(3, -1), D(3, -4) and E(1, -6).

Also given a graph of the polygon A'B'C'D'E'.

A'(13, -4), B'(13, -1), C'(9, -1), D'(9, -4) and E'(11, -6).

Here the y coordinates has not been change. Only the x coordinates change.

So the line of reflection will be of the form x = c.

Mid point of each of the corresponding vertices are,

Mid point of AA' = (6, -4),

Midpoint of BB' = (6, -1)

Midpoint of CC' = (6, -1)

Midpoint of DD' = (6, -4)

Midpoint of EE' = (6, -6)

So all the x coordinates are same, x = 6

Hence the line of reflection is x = 6.

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From the given triangle figure the value of y is 4.5.

The given figure is -

Now we have to find the value of 'y' from the given triangle.

Triangle ABC is right angled triangle with right angle at point A.

So, ∠B + ∠C = 90° ......... (i)

Again, triangle ADB is a right angled triangle with right angle at point D.

So, ∠DAB + ∠B = 90° .......... (ii)

Comparing (i) and (ii) we can say that,

∠DAB = ∠C

Again, triangle ADB is a right angled triangle with hypotenuse AB.

By Pythagoras theorem,

AD² + DB² = AB²

AB² = 3² + 2² = 9 + 4 = 13

AB = √13

Hence triangle ABC and triangle ADB are two similar triangles. So,

AB/DB = CB/AB

√13/2 = (y + 2)/√13

2(y + 2) = √13*√13

2y + 4 = 13

2y = 13 - 4 = 9

y = 9/2 = 4.5

So the value of y is 4.5.

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The question in another language, The question in English should be -

"In the figure below, find the exact value of y. (without making an approximation of the answer)"

Answer:

Step-by-step explanation:

a. f(-a) = 3 - 2(-a) = 3 + 2a

b. -f(a) = -(3 - 2a) = -3 + 2a

c. f(a + h) = 3 - 2(a + h) = 3 - 2a - 2h

d. f(a) + f(h) = (3 - 2a) + (3 - 2h) = 6 - 2a - 2h

e. (f(a + h) - f(a))/h = [3 - 2(a + h) - (3 - 2a)]/h = [-2h]/h = -2, if h ≠ 0.

Answer:

a. To find f(-a), substitute -a for x in the given function:

f(-a) = 3 - 2(-a) = 3 + 2a

Therefore, f(-a) = 3 + 2a.

b. To find -f(a), evaluate f(a) and then multiply by -1:

f(a) = 3 - 2a

-f(a) = -(3 - 2a) = -3 + 2a

Therefore, -f(a) = -3 + 2a.

c. To find f(a + h), substitute a + h for x in the given function:

f(a + h) = 3 - 2(a + h) = 3 - 2a - 2h

Therefore, f(a + h) = 3 - 2a - 2h.

d. To find f(a) + f(h), evaluate f(a) and f(h), and then add them:

f(a) = 3 - 2a

f(h) = 3 - 2h

f(a) + f(h) = (3 - 2a) + (3 - 2h) = 6 - 2a - 2h

Therefore, f(a) + f(h) = 6 - 2a - 2h.

e. To find (f(a + h) - f(a))/h if h ≠ 0, substitute the expressions for f(a + h) and f(a) into the formula:

(f(a + h) - f(a))/h = ((3 - 2a - 2h) - (3 - 2a))/h = (-2h)/h = -2

Therefore, (f(a + h) - f(a))/h = -2 if h ≠ 0.

The prediction that is not supported by the data is option B: "The number of non-fiction books sold on Friday will be two-and-a-half times the number of arts and crafts books sold on Friday."

We can see from the table that on Thursday, 7 sports and trivia books and 19 arts and crafts books were sold, for a difference of 12.

On Thursday, 13 non-fiction books and 19 arts and crafts books were sold.

If we assume that the same ratio will hold on Friday, then we can predict that the number of non-fiction books sold will be (19/2)×2.5 = 23.75 Which is not a whole number.

Therefore, this prediction is not supported by the data.

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