Nadya has 36 blueberries and 45 raspberries that she will be putting into cups of yogurt to make after-school snacks. She wants to put an equal number of blueberries and an equal number of raspberries into each cup. She uses all the berries.

Answer:

Area of the field is 540 m².

Step-by-step explanation:

ABCD is the given quadrilateral in which diagonal BD is 36 m.

Now, AL ⊥ BD and CM ⊥ BD. Also, AL = 19 m and CM = 11 m.

Now, we have to calculate the area of quadrilateral shaped field ABCD.

At first, we will find the area of ΔABD and ΔBCD and then we will add the area of both the triangles to get the area of the quadrilateral shaped field.

Now, ΔABD and ΔBCD are both right angled triangles.

So,

[tex]area\; of \; triangle \; ABD = \frac{1}{2} \times base\times height[/tex]

[tex]=\frac{1}{2}\times BD\times AL=\frac{1}{2}\times36\times19=342\; m^{2}[/tex]

[tex]area \; of \; triangle\; BCD = \frac{1}{2}\times BD\times CM=\frac{1}{2}\times36\times 11 = 198\; m^{2}[/tex]

So, area of field ABCD = area of ΔABD + area of ΔCBD

= 342 + 198

= 540 m²

So, the area of quadrilateral shaped field is 540 m².

Answer:

The numbers are 23,27

Step-by-step explanation:

Let one number be x

Other number = 50 - x

50 - x - x = 4

50 - 2x = 4

-2x = 4 - 50

-2x = - 46

x = -46/-2

x = 23

Other number = 50 - x = 50 -23 = 27

Answer:

1275

Step-by-step explanation:

This is an arithmetic series. The formula for this is Sₙ = (n/2)(a₁ +aₙ)d. a₁ is the first term, so here it is 1, and aₙ is the nth term, or the last term, which is 50 here, but we don't know n.

Now we have to use the equation for an arithmetic sequence to solve for n. A sequence would just be if there was not a last number and it went on forever. that equation looks like aₙ = a₁ + (n - 1)d. Now the only new variable is d, which is the common difference. You can find that by subtracting one term from the term before it, like 2-1 = 1, so d is 1.

We can now solve for n by plugging our numbers into the second equation, so 50 = 1 + (n - 1)1, we can distribute the 1 and to (n-1) and get 50 = 1 + n - 1. Now the ones will cancel and we are left with n = 50

Finally we can plug everything into our original equation and find Sₙ = (50/2)(1+50), which simplifies to Sₙ = 25(51), and Sₙ = 1275.

The width of rectangle is 8 units.

Step-by-step explanation:

Given,

Area of rectangle = 40 units

Width = w

Length = w-3

Area = Length * Width

[tex]40=(w-3)*w\\40=w^2-3w\\w^2-3w=40\\w^2-3w-40=0[/tex]

Factorizing the equation

[tex]w^2-8w+5w-40=0\\w(w-8)+5(w-8)=0\\(w-8)(w+5)=0[/tex]

Either,

w-8=0         => w=8

Or,

w+5=0        =>w= -5

As width cannot be negative, therefore

Width of rectangle = 8 units

The width of rectangle is 8 units.

Keywords: area, rectangle

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Final answer:

To find the width of a rectangle with an area of 40 units and a length expressed as the width minus 3 units, we set up and solve a quadratic equation, yielding the width of the rectangle as 8 units.

Explanation:

The question asks to find the width of a rectangle when the length is the width minus 3 units and the area is 40 units. To solve this, let's denote the width as w, then the length will be w - 3. The area of a rectangle is calculated by multiplying the length by the width, so we will set up an equation: area = length × width, or 40 = w × (w - 3).

This is a quadratic equation: w² - 3w - 40 = 0. To solve it we can factor the quadratic or use the quadratic formula. Factoring gives us (w - 8)(w + 5) = 0, which means w could be 8 or -5. Since a width cannot be negative, the width of the rectangle is 8 units.

Answer: y = x - 9

Step-by-step explanation:

The equation of line slope - point form is given as :

y - [tex]y_{1}[/tex] = m ( x -[tex]x_{1}[/tex]

From the question

m = 1

[tex]x_{1}[/tex] = 9

[tex]y_{1}[/tex] = 0

Substituting into the formula , we have

y - 0 = 1 (x - 9)

y = x - 9

Therefore , the equation of the line in slope - intercept form is given as

y = x - 9

The equation of the line in slope-intercept form is y = x - 9.

The equation of a line in slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept. Given that the slope of the line is 1 and it passes through the point (9, 0), we can substitute the values into the equation.

Let's substitute the slope m = 1 and the x-coordinate of the point x = 9:

y = 1(9) + b

Since the point (9, 0) lies on the line, we can substitute the y-coordinate y = 0:

0 = 9 + b

Solving for b, we subtract 9 from both sides:

b = -9

Therefore, the equation of the line in slope-intercept form is y = x - 9.

Answer:

maximum height reached = 35 feet

and [tex]f(t) = 48t-16.07t^{2}[/tex]

Step-by-step explanation:

writing linear motion equations

[tex]s = ut + \frac{1}{2}at^{2}[/tex]

where s is the total displacement, u the initial velocity, t the time travelled, and a is the acceleration.

given u = 48 ft/s, and a = acceleration due to gravity g = -9.8[tex]\frac{m}{s^{2}}[/tex]

1 m = 3.28 feet therefore g becomes -9.8×3.28[tex]\frac{ft}{s^{2}}[/tex]

here negative sigh comes as acceleration due to gravity is in opposite direction of initial velocity.

therefore f(t) becomes [tex]f(t) = 48t-16.07t^{2}[/tex]

to find max height we should find differentiation of f(t) and equate it to 0

therefore we get 48 = 32.144t

t = 1.49 s

therefore max height f(1.49) = 71.67-36.67 = 35 feet

Answer: 36!!!

Step-by-step explanation:

The cost of swimming lesson in winters is $60.

Step-by-step explanation:

Given,

Cost of lessons in summer = $50 per month

Raise in winter = 20%

Amount of raise = 20% of summer's cost

Amount of raise = [tex]\frac{20}{100}*50[/tex]

Amount of raise = [tex]\frac{1000}{100}=\$10[/tex]

Cost of lesson in winter = Cost in summer + Amount of raise

Cost of lesson in winter = 50+10 = $60

The cost of swimming lesson in winters is $60.

Keywords: percentage, addition

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Answer:

4/9.

Step-by-step explanation:

There are a total of 18 people.

Prob( A Sculptor is chosen) = 3/18 = 1/6.

Prob( a photographer) = 5/18

The required probability is the sum of these 2, so

it is 3/18 + 5/18

= 8/18

= 4/9.

Answer:

[tex]\frac{8}{17}[/tex]

Step-by-step explanation:

Answer:

Therefore the required Equations are

[tex]y =2(x+3)\ \textrm{is the required expression for First condition.}\\y = 3x-2\ \textrm{is the required expression for Second condition.}[/tex]

Step-by-step explanation:

Given:

'y' represents the larger number and

'x' represents the smaller number

Then, a larger number is double the sum of 3 and a smaller number will be

Larger no = double of ( 3 and smaller number)

∴ [tex]y=2(x+3)\ \textrm{is the required expression for First condition.}[/tex]

Now,

The larger number is 2 less than 3 times the smaller number.

Larger number = 3 times smaller number and 2 less

∴ [tex]y=3x-2\ \textrm{is the required expression for Second condition.}[/tex]

Therefore the required Equations are

[tex]y =2(x+3)\ \textrm{is the required expression for First condition.}\\y = 3x-2\ \textrm{is the required expression for Second condition.}[/tex]

Final answer:

The correct equations that model the situation are y = 3x - 2 and y = 2(x + 3), as they reflect the two conditions given in the problem description.

Explanation:

To determine which equations model the situation described, we should translate the worded statements into algebraic equations. The first statement tells us that a larger number 'y' is double the sum of 3 and a smaller number 'x'. This can be written as y = 2(x + 3). The second statement tells us that the larger number is 2 less than 3 times the smaller number, which can be written as y = 3x - 2.

Now, we need to verify the provided choices against these two derived equations:

y = 3x - 2 (Correct, it matches the second statement we translated from the problem description.)

3x - y = 3 (Incorrect, because rearranging this gives y = 3x - 3, which does not match either of our derived equations.)

3x - y = -2 (Incorrect, because rearranging this gives y = 3x + 2, which does not match either of our derived equations.)

y = 2 - 3x (Incorrect, this does not match the format of either derived equation.)

y=2(x + 3) (Correct, it matches the first statement we translated from the problem description.)

Final answer:

The number where the value of the digit 6 is 10 times the value in the number 186,425 is 462,017, as the 6 is in the ten thousands place, giving it a value of 60,000. The correct answer is option B.

Explanation:

The student is tasked with finding the place value of 6 that is 10 times the value of the 6 in the number 186,425. In 186,425, the 6 is in the thousands place, so its value is 6,000 (6 x 103). Therefore, to find the number where the value of 6 is ten times 6,000, we need a 6 that is worth 60,000. The 6 must be in the ten thousands place to have this value.

Let's inspect the options:

A. 681,452 - Here, the 6 is in the hundred thousands place, so the value of 6 is actually 600,000, which is not 10 times 6,000.

B. 462,017 - The 6 is in the ten thousands place. Here, the value of 6 is 60,000, which is 10 times the value of 6 in the number 186,425.

C. 246,412 - The 6 is in the thousands place again, so the value is 6,000, not 10 times more.

D. 125,655 - The 6 here does not multiply its value since it's in the tens place.

Therefore, the correct answer is option B, where the value of the digit 6 is exactly 10 times the value of the digit 6 in the number Andrew wrote.

Answer:

Option C is true.

Step-by-step explanation:

See the attached diagram.

Since MN is a diameter of the circle at P, so it will always make a right angle at a point on the circumference of the circle.

Therefore, ∠ MLN = 90° as L is a point on the circumference.

Now, given that LM = 2x and LN = 3x, then we have to find MN.

So, applying Pythagoras Theorem, MN² = LM² + LN²

⇒ MN² = 4x² + 9x² = 13x²

⇒ MN = x√13

Therefore, option C is true. (Answer)

Answer:

[tex]a=36.87\ units[/tex]

[tex]B=57.47^o[/tex]

[tex]C=71.53^o[/tex]

Step-by-step explanation:

step 1

Find the length side a

Applying the law of cosines

[tex]a^2=b^2+c^2-2(b)(c)cos(A)[/tex]

substitute the given values

[tex]a^2=40^2+45^2-2(40)(45)cos(51^o)[/tex]

[tex]a^2=1,359.4466[/tex]

[tex]a=36.87\ units[/tex]

step 2

Find the measure of angle B

Applying the law of sines

[tex]\frac{a}{sin(A)} =\frac{b}{sin(B)}[/tex]

substitute the given values

[tex]\frac{36.87}{sin(51^o)} =\frac{40}{sin(B)}[/tex]

[tex]sin(B)=\frac{sin(51^o)}{36.87}{40}[/tex]

[tex]B=sin^{-1}(\frac{sin(51^o)}{36.87}{40})=57.47^o[/tex]

step 3

Find the measure of angle C

Remember that the sum of the interior angles in any triangle must be equal to 180 degrees

so

[tex]A+B+C=180^o[/tex]

substitute the given values

[tex]51^o+57.47^o+C=180^o[/tex]

[tex]108.47^o+C=180^o[/tex]

[tex]C=180^o-108.47^o=71.53^o[/tex]

Answer:

The rate at which the value of house increases in 40 years is 1.03  

Step-by-step explanation:

The initial value of house = P = $30,000

The final value of house = F = $120,000

The period for which the value increase = 40 years

Let the rate at which the value increases in 40 years = r%

Now, According to question

The final value of house after n years = The initial value of house × [tex](rate)^{time}[/tex]

i.r F = P × [tex](r)^{n}[/tex]

Or, r = [tex](\frac{F}{P})^{\frac{1}{n}}[/tex]

Or, r = [tex](\frac{120,000}{30,000})^{\frac{1}{40}}[/tex]

Or, r = [tex]4^{\frac{1}{40}}[/tex]

∴  r = 1.03

The rate at which the value increases in 40 years = r = 1.03

Hence,The rate at which the value of house increases in 40 years is 1.03  Answer

Answer:

The answer is 305. Step 1 is wrong in the given answer.

Kevin's mistake is that his step 1 in the answer is  wrong.

Step-by-step explanation:

Given expression is [tex]\frac{2440}{8}[/tex]

Now to simplify the given expression:

Given expression can be written as below

Step 1: [tex]\frac{2440}{8}=\frac{2400+40}{8}[/tex]

Step 2:  [tex]\frac{2440}{8}=\frac{2400}{8}+\frac{40}{8}[/tex]

Step 3: [tex]\frac{2400}{8}+\frac{40}{8}=300+5[/tex]  (the sum of the numerators are dividing with their corresponding terms)

Step 4: [tex]\frac{2440}{8}=305[/tex]  (adding the terms)

Step 5: Therefore [tex]\frac{2440}{8}=305[/tex]

Therefore the answer is 305 and from the problem step 1 is wrong.

Kevin's mistake is that his answer is correct and the step 1 is  wrong.

The answer is 305. Step 1 is wrong in the given answer.

Answer:

2496in3

Step-by-step explanation:

12x16x13=2496in3

Answer:

2496 in³

Step-by-step explanation:

refer to attached graphic

Given:

Length,l = 12 in

height,h = 16 in

width,w = 13 in

Volume,

= lwh

= 12 x 13 x 16

=  2496 in³

Answer: 3.82 miles

Step-by-step explanation:

According to the shown figure, we can imagine the scene as a right triangle, where the tower is located at the right angle, and the fireman and the forest fire located at each of the other two vertices.

So, since we are dealing with a right triangle we can use the Pithagorean Theorem, in order to find the distance from the fireman to the fire [tex]d[/tex], which is also the hypotenuse.

[tex]d^{2}= (2.1 miles)^{2} +(3.2 miles)^{2}[/tex]

[tex]d=\sqrt{(2.1 miles)^{2} +(3.2 miles)^{2}}[/tex]

Finally:

[tex]d=3.82 miles[/tex]

The distance from the fireman and the fire is 3.83 miles.

Using the values given, we can calculate the distance using pythagoras ;

Inputting the values into the formula ;

distance = √2.1² + 3.2²

distance = √14.65

distance = 3.8275318418

Therefore, the distance between the fireman and the fire is 3.83 miles.

Final answer:

To find the actual height of the door based on a scale model, the scale factor between the model and the actual house (1:36) is used, leading to the conclusion that the real door is 6 feet high.

Explanation:

The question involves scale and measurement to find the actual height of the door of a real house based on its scale model. The scale model of the house is 1 foot long, and the actual house is 36 feet long. The door in the model is 2 inches high. To find the actual height of the door, we use the scale factor between the model and the real house.

First, we identify the scale factor: Since the model house is 1 foot long and the actual house is 36 feet long, the scale factor is 1:36. Next, we convert the height of the door from inches to feet in the model scale (since 1 foot = 12 inches, 2 inches = 1/6 feet). Using the scale factor, the height of the actual door is calculated as follows:

Height in the model (in feet) x Scale factor = Height of the actual door

1/6 feet x 36 = 6 feet

Therefore, the actual height of the door is 6 feet.

The height of the actual door is 6 feet, calculated using the scale factor of 1:36 from the 2-inch model door height.

To find the height of the actual door from the scale model measurements, we first need to determine the scale factor between the model and the real house. The scale model is 1 foot long, and the actual house is 36 feet long, which means that the scale factor is 1:36. This implies that every inch on the model would represent 36 inches (or 3 feet) on the actual house. Since the door on the model is 2 inches high, we can calculate the height of the actual door by multiplying the model door height (2 inches) by the scale factor.

So, height of actual door = 2 inches * 36 inches/inch = 72 inches.

To convert 72 inches to feet, we divide by 12, since there are 12 inches in a foot.

Therefore, height of actual door in feet = 72 inches / 12 inches/foot = 6 feet.

The actual door is 6 feet high.

Answer:

The third option is the correct one.

Step-by-step explanation:

For similar triangles, the angles are the same, but the distances between corresponding vertices is only proportional

Answer:

12/25

Step-by-step explanation:

2/5 ÷ 5/6

To divide by a fraction, multiply by the reciprocal.

2/5 × 6/5

12/25

Answer:

To write 252 cards together Courtney and Victoria will take = 140 minutes or 2 hours and 20 minutes.

Step-by-step explanation:

Given:

Courtney writes 4 cards in 10 minutes

Victoria writes 14 cards in 10 minutes.

To find the time taken by them to  write 252 cards working together.

Solution:

Using unitary method to determine their 1 minute work.

In 10 minutes Courtney writes = 4 cards

So,in 1 minute Courtney will write = [tex]\frac{4}{10}[/tex] cards

In 10 minutes Victoria writes = 14 cards

So,in 1 minute Victoria will write = [tex]\frac{14}{10}[/tex] cards

Now, Courtney and Victoria are working together.

So, in 1 minute, number of cards they can write together will be given as:

⇒ [tex]\frac{4}{10}+\frac{14}{10}[/tex]

Since we have common denominators, so we can simply add the numerators.

⇒ [tex]\frac{18}{10}[/tex]

Again using unitary method to determine the time taken by them working together to write 252 cards.

They can write  [tex]\frac{18}{10}[/tex] cards in 1 minute.

To write 1 card they will take  = [tex]\frac{1}{\frac{18}{10}}=\frac{10}{18}[/tex] minutes

So, for 252 cards the will take = [tex]\frac{10}{18}\times252=\frac{2520}{18}=140[/tex] minutes

140 minutes = (60+60+20) minutes = 2 hours and 20 minutes [ As 60 minutes = 1 hour]

So, to write 252 cards together Courtney and Victoria will take = 140 minutes or 2 hours and 20 minutes.

Courtney and Victoria can write 252 Christmas cards in 140 minutes when working together.

To find out how long it will take Courtney and Victoria to write 252 Christmas cards when they work together, we need to first determine how many cards they can write in 10 minutes individually. Courtney can write 4 cards in 10 minutes, while Victoria can write 14 cards in the same time. Thus, Courtney can write 4/10 = 0.4 cards per minute, and Victoria can write 14/10 = 1.4 cards per minute. When they work together, their combined rate is 0.4 + 1.4 = 1.8 cards per minute.

To determine the total time it will take them to write 252 cards, we can divide the number of cards by their combined rate: 252 / 1.8 = 140 minutes. Therefore, it will take them 140 minutes to write out 252 Christmas cards when they work together.

Answer:

[tex]\displaystyle about\:8\:units[/tex]

Step-by-step explanation:

Use the Distance Formula:

[tex]\displaystyle \sqrt{[-x_1 + x_2]^2 + [-y_1 + y_2]^2} = D \\ \\ \sqrt{[6 - 1]^2 + [-7 + 1]^2} = \sqrt{5^2 + [-6]^2} = \sqrt{25 + 36} = \sqrt{61} ≈ 7,810249676 ≈ 8[/tex]

Since we are talking about distance, we ONLY want the NON-NEGATIVE root.

I am joyous to assist you anytime.

Final answer:

To find (g - f)(3) with given functions f(x) and g(x), subtract f(x) from g(x) to get the new function, then evaluate this function at x = 3. The result is 23.

Explanation:

The student is asking to find the result of the operation (g - f)(3) where f(x) = 4 - x2 and g(x) = 6x. (g - f)(x) means we subtract the function f from the function g, and then evaluate the resulting function at x = 3.

To solve this:

So (g - f)(3) is 23.

                                   Question #15

Step-by-step explanation:

A notation such as [tex]T_{(-1, 1)}oR_{y-axis}[/tex] is read as:

"a translation of (x, y) → (x - 1, y + 1) after a reflection across y-axis.

The rule of reflection of point (x, y) across y-axis brings (x, y) → (-x, y), meaning that y-coordinate remains the same, but x-coordinate changes its sign.

As ΔABC with coordinates A(1, 3), B(4, 5) and C(5, 2). Here is the coordinates of ΔA'B'C' after the glide reflection described by [tex]T_{(-1, 1)}oR_{y-axis}[/tex].

                                            [tex]R_{y-axis}[/tex]                                   [tex]T_{(-1, 1)}[/tex]  

A(1, 3)              →                A'(-1, 3)               →                    A"'(-2, 4)

B(4, 5)             →                B'(-4, 5)               →                    B"'(-5, 6)

C(5, 2)             →                C'(-5, 2)               →                    C"'(-6, 3)

                                        Question #16

Step-by-step explanation:

A glide reflection is said to be a transformation that involves a  translation followed by a reflection in which every  point P is mapped to a point P ″ by the following steps.

As ΔABC with coordinates A(-4, -2), B(-2, 6) and C(4, 4).

Translation : (x, y)  → (x + 2, y + 4)

Reflection : in the x-axis

The rule of reflection of point (x, y) across x-axis brings (x, y) → (x, -y), meaning that x-coordinate remains the same, but y-coordinate changes its sign.

Hence,

ΔABC with coordinates A(-4, -2), B(-2, 6), C(4, 4) after (x, y)  → (x + 2, y + 4) and reflection in the x-axis.

A(-4, -2)         →     A'(-2, 2)     →     A''(-2, -2)    

B(-2, 6)          →    B'(0, 10)       →     B''(0, -10)

C(4, 4)            →    C(6, 8)         →      C''(6, -8)

Keywords: reflection, glide reflection, translation

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Answer:

A) [tex]y=\frac{1}{2}x+3[/tex] - Linear

B) [tex]y=4x+2[/tex] - Linear      

C) [tex]xy=12[/tex] - Nonlinear

Step-by-step explanation:

To determine whether a function is linear or nonlinear.

The function of a straight line is given as :

[tex]y=mx+b[/tex]

where [tex]m[/tex] represents slope of line and [tex]b[/tex] represents the y-intercept.

Any function that can be represented as a function of straight line is called a linear function otherwise it is nonlinear.

We will check the equations given for linear or nonlinear.

A) [tex]y=\frac{1}{2}x+3[/tex]

The function is in the form [tex]y=mx+b[/tex] and hence it is a linear function with slope [tex]m=\frac{1}{2}[/tex] and y-intercept [tex]b=3[/tex].

B) [tex]y=4x+2[/tex]

The function is in the form [tex]y=mx+b[/tex] and hence it is a linear function with slope [tex]m=4[/tex] and y-intercept [tex]b=2[/tex].

C) [tex]xy=12[/tex]

On solving for [tex]y[/tex]

Dividing both sides by  [tex]x[/tex]

[tex]\frac{xy}{x}=\frac{12}{x}[/tex]

[tex]y=\frac{12}{x}[/tex]

This function cannot be represented in the form [tex]y=mx+b[/tex], hence it is a nonlinear function.

Answer:

A) H + R + T = 109

B) R = 3T

C) T = H + 9 combining equation B) with equation A)

A) H + 4T = 109 combining C) with A)

A) T -9  + 4T = 109

A) 5T = 118

Tom has 23.60 dollars

We put this information into equation B)

R = 3*23.60

Rafeal has 70.80 dollars

Putting this into Rafael and Ton into equation A)

A) H + 70.80 + 23.60 = 109

Heather has 14.60 dollars

Step-by-step explanation:

We are Given:

Height of the Pyramid(h) = 2 cm

Volume of the Pyramid = 24 cm³

Base of the Pyramid:

We know that the Volume of a square-based Pyramid:

Volume = a²*(h/3)

24 = a² * (2/3)                 [Replacing the variables]

24 * 3/2 = a²                   [Multiplying both sides by 3/2]

a² = 36

a = 6                               [taking the square root of both sides]

Hence, the length of base of the Pyramid is 6 cm

Final answer:

The length of the base of the square pyramid is 6 centimeters, which is found by solving the equation V = (1/3) * B * h for the area of the base B and then finding the square root of B to get the length of the side.

Explanation:

To determine the length of the base of the square pyramid, we can use the formula for the volume of a pyramid, which is V = (1/3) * B * h, where V is the volume, B is the area of the base, and h is the height of the pyramid. For a square pyramid, the base is a square, so the area of the base B can be expressed as s2, where s is the length of the side of the square.

We are given that the volume V of the pyramid is 24 cm³ and the height h is 2 cm. Using the formula, we can solve for s:

V = (1/3) * s2 * h

24 cm³ = (1/3)  * s^2 * 2 cm

24 cm³ = (2/3) * s^2

s^2 = (24 * 3) / 2

s^2 = 36

s =[tex]\sqrt{36}[/tex]

s = 6 cm

Therefore, the length of the base of the square pyramid is 6 centimeters.

Answer:

x = 11

m∠B = 80°

Step-by-step explanation:

If two parallel lines are cut by a transversal, the corresponding angles are congruent

m∠A = m∠B

8x - 8 = 5x + 25 ... minus 5x and add 8 both side

8x - 5x = 25 + 8

3x = 33

x = 11

m∠B = 5 x 11 + 25 = 80°

check: m∠A = 8 x 11 -8 = 80

Answer:

Part a) The solution is the ordered pair (6,10)

Part b) The solutions are the ordered pairs (7,3) and (15,1.4)

Step-by-step explanation:

Part a) we have

[tex]\frac{x}{2}-\frac{y}{5}=1[/tex] ----> equation A

[tex]y-\frac{x}{3}=8[/tex] ----> equation B

Multiply equation A by 10 both sides to remove the fractions

[tex]5x-2y=10[/tex] ----> equation C

isolate the variable y in equation B

[tex]y=\frac{x}{3}+8[/tex] ----> equation D

we have the system of equations

[tex]5x-2y=10[/tex] ----> equation C

[tex]y=\frac{x}{3}+8[/tex] ----> equation D

Solve the system by substitution

substitute equation D in equation C

[tex]5x-2(\frac{x}{3}+8)=10[/tex]

solve for x

[tex]5x-\frac{2x}{3}-16=10[/tex]

Multiply by 3 both sides

[tex]15x-2x-48=30[/tex]

[tex]15x-2x=48+30[/tex]

Combine like terms

[tex]13x=78[/tex]

[tex]x=6[/tex]

Find the value of y

[tex]y=\frac{x}{3}+8[/tex]

[tex]y=\frac{6}{3}+8[/tex]

[tex]y=10[/tex]

The solution is the ordered pair (6,10)

Part b) we have

[tex]xy=21[/tex] ---> equation A

[tex]x+5y=22[/tex] ----> equation B

isolate the variable x in the equation B

[tex]x=22-5y[/tex] ----> equation C

substitute equation C in equation A

[tex](22-5y)y=21[/tex]

solve for y

[tex]22y-5y^2=21[/tex]

[tex]5y^2-22y+21=0[/tex]

Solve the quadratic equation by graphing

The solutions are y=1.4, y=3

see the attached figure

Find the values of x

For y=1.4

[tex]x=22-5(1.4)=15[/tex]

For y=3

[tex]x=22-5(3)=7[/tex]

therefore

The solutions are the ordered pairs (7,3) and (15,1.4)

Answer:

Option C is correct.

Step-by-step explanation:

See the diagram attached.

Given that YZ bisects MO, hence, MZ = ZO ........ (1)

If we want to prove that point N is equidistant from points M and O, then we have to prove that Δ MNZ ≅ Δ ONZ, so that we can prove that MN = ON.

Now, to prove Δ MNZ ≅ Δ ONZ, we must have another condition that MO ⊥ YZ or, NZ ⊥ MO.

So, we have (i) MZ = OZ {from equation (1)}

(ii) ∠ NZM = ∠ NZO = 90° {Since, NZ ⊥ MO} and  

(iii) NZ is the common side

Hence, by SAS criteria it is proved that Δ MNZ ≅ Δ ONZ and hence, proved that MN = ON.

Therefore, option C is correct. (Answer)