Bookwork code: K27
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allowed
A pendulum of length 18 cm is swinging from a fixed point, as shown.
Calculate the distance that the end of the pendulum travels when it swings
through an angle of 87°.
Give your answer in centimetres (cm) to 1 d.p.
87°
18 cm

The height of the tree stump is approximately 32.5 inches. Where its radius as 22 in and its volume as 49265 cubic inches.

The volume of a right cylinder is calculated as follows:

V = πr²h

where V denotes volume, r denotes radius, and h denotes height.

We are provided the radius and volume of the tree stump in this scenario. We may plug these values into the algorithm to find the height:

49265 = π(22²)h

Simplifying the right-hand side, we get:

49265 = 484πh

Dividing both sides by 484π, we get:

h = 49265 / (484π)

h = 49265 / (484 * 3.14)

h = 49265/ 1519.76

h = 32.416

h ≈ 32.5

Therefore, the height of the tree stump is approximately 32.5 inches.

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Answer:

[tex]\sqrt{844}=2\sqrt{211}=29.05[/tex]

[tex]\sqrt{900} =30[/tex]

[tex]\sqrt{650}=5\sqrt{26}=25.49[/tex]

[tex]\sqrt{784}=28[/tex]

[tex]\sqrt{699} =\sqrt{699}=26.43[/tex]

[tex]\sqrt{805}=\sqrt{805}=28.37[/tex]

[tex]\sqrt{729}=27[/tex]

Step-by-step explanation:

1.

[tex]\sqrt{844}[/tex]

[tex]\sqrt{2} *\sqrt{422}[/tex]

[tex]1\sqrt{422}[/tex]

[tex]1\sqrt{2} *\sqrt{211}[/tex]

[tex]1+1\sqrt{211}[/tex]

[tex]2\sqrt{211}[/tex]

[tex]29.05[/tex]

2.

[tex]\sqrt{900}[/tex]

[tex]\sqrt{9} *\sqrt{100}[/tex]

[tex]3\sqrt{100}[/tex]

[tex]3 \sqrt{25} *\sqrt{4}[/tex]

[tex]3*5*2[/tex]

[tex]30[/tex]

3.

[tex]\sqrt{650}[/tex]

[tex]\sqrt{25}*\sqrt{26}[/tex]

[tex]5\sqrt{26}[/tex]

[tex]25.49[/tex]

4.

[tex]\sqrt{784}[/tex]

[tex]\sqrt{4} *\sqrt{196}[/tex]

[tex]2\sqrt{196}[/tex]

[tex]2\sqrt{49}*\sqrt{4}[/tex]

[tex]2*7*2[/tex]

[tex]28[/tex]

5.

[tex]\sqrt{699}[/tex]

[tex]26.43[/tex]

(can't be simplified)

6.

[tex]\sqrt{805}[/tex]

[tex]28.37[/tex]

(Can't be simplified)

7.

[tex]\sqrt{729}[/tex]

[tex]\sqrt{9}*\sqrt{81}[/tex]

[tex]3*9[/tex]

[tex]27[/tex]

Answer:

D

Step-by-step explanation:

First, to find the length of the side FH we will use the Pythagorean theorem which states [tex]a^2+b^2=c^2[/tex]. We can say that FH = a, FG = b, and HG = c. Since we know the values of b and c, we will rearrange the formula to solve for a [tex]a = \sqrt{c^2-b^2}[/tex]. That will be [tex]\sqrt{18^2-12^2)[/tex] which gives us 13.4 for the missing side.

Now to solve for the angles we will use simple trigonometry. Since we know that [tex]sin(\theta) = \frac{opposite} {hypotenuse}[/tex], we can do inverse trigonometry for the missing angles [tex]sin^{-1}(\frac{opposite}{adjacent}) = \theta[/tex]. For angle H, we can do [tex]sin^{-1}(\frac{12}{18})[/tex] which gives us 41.8 degrees. For angle G we can do [tex]sin^{-1}(\frac{13.4}{18})[/tex] and we get 48.1 degrees. We could also use inverse cos and trig to solve it.

Looking at the answer choices, only answer choice D is correct.

The correct graph of the inequality ( |x| + 3 > 3) is attached accordingly.

Recall that the absolute value (or modulus) of a real number x is its non-negative value regardless of its sign. For example, 5 has an absolute value of 5, and 5 has an absolute value of 5.

A number's absolute value can be conceived of as its distance from zero along the real number line.

So, given  |x| + 3 > 3

Add minus three to both sides

|x| + 3  + (-3) > 3 + (-3)

|x|  > 0

Thus, the absolute value of x is

x > 0 or x < -0

put in the form of inequality, we have

x > 0 or x < 0

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The distance between the points are matched as;

1. 10. 6

2. 7. 6

3. 7. 1

Using the Pythagorean theorem, we have that the square of the hypotenuse side is equal to the sum of the squares of the other two sides, we have that;

1. Opposite = 8

Adjacent = 7

Substitute the values

x² =8² + 7²

Find the square

x² = 64 + 49

x = √113

x = 10. 6

2. x²= 3² + 7²

Find the square value

x² = 9 + 49

x² = 58

Find the square root

x = 7. 6

3. x² = 5² + 5²

find the value

x² = 50

x = √50

x = 7. 1

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The additional interest on the loan by the rates is $2150241.13

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

Principal, P = $271,000

Time = 40 years

Rates = 7% and 5%

The interest is calculated as

I = P(1 + r)ˣ - P

So, we have

Calculate the difference in the interests

Difference = 3787078.07443 - 1636836.94099

So, we have

Difference = 2150241.13344

Approximate

Difference = 2150241.13

Hence, the additional interest is $2150241.13

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Answer:

0.3947368421 = approx 39%

Step-by-step explanation:

Add up all the number of times rolled on the right column.

=6+3+5+6+9+9

= 38 total rolls

Add up the number of times a 4 or a 5 were rolled:

= 6+ 9 = 15

Divide 15/38

=15/38=0.3947368421 = approx 39%.

Answer: 350000

Step-by-step explanation: Let the total sale of noodles by the wholesaler be x.

Given: Commission for the wholesaler is 10%

After receiving a 10% commission the company got rupees 315000.

x-10% of x =315000

x - 10\100 = 315000

x-x\10 = 315000

10x - 10\10 =315000

9x = 3150000

x= 3150000\9

x= 350000

The partial fraction of the given rational expression is [tex]\frac{25}{(x+1(x+2)} =\frac{25A}{3(x-1)} -\frac{25B}{3(x+2)}[/tex]

Given the following equation below:

[tex]\frac{25}{(x+1(x+2)}[/tex]

We are expression as the sum of partial fractions

Let the numerator of both partial expressions be A and B to have:

[tex]\frac{25}{(x+1(x+2)} =\frac{A}{x-1} +\frac{B}{x+2}[/tex]

Simplifying the expression, we will have;

[tex]\frac{25}{(x+1(x+2)} =\frac{A(x+2)+B(x-1)}{x-1(x+2)}[/tex]

Cancelling the denominator:

25 = A(x+2) + B(x-1)


If x - 1 = 0, then x = 1. Substitute to have:

25 = A(1+2)

3A = 25
A = 25/3

If x + 2 = 0, x = -2

25 = B(-3)
B = -25/3

Substitute into the function to have:

[tex]\frac{25}{(x+1(x+2)} =\frac{25A}{3(x-1)} -\frac{25B}{3(x+2)}[/tex]

This gives the partial fraction of the given rational expression.

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Answer:

The total number of marbles in the bag is 4 + 12 + 8 + 2 = 26.

The probability of not picking a red marble out of the bag is equal to the number of marbles that are not red divided by the total number of marbles in the bag.

The number of marbles that are not red is 4 (blue) + 8 (green) + 2 (yellow) = 14.

Therefore, the probability of not picking a red marble out of the bag is 14/26 or 7/13.

So the probability of not picking a red marble is 7/13.

The common difference is -3, and the terms are 17, 14, 11, 8, 5 and the formula is a(n) = 17 - 3(n - 1)

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

a20 = -40

a37 = -91

An arithmetic sequence is represented as

a(n) = a + (n - 1) * d

Where

d = common difference

Using the given terms, we have

a + 19d = -40

a + 36d = -91

So, we have

17d = -51

Divide

d = -3

This also means that

a + 19(-3) = -40

So, we have

a  = 17

Calculating the first five terms, we have

17, 14, 11, 8, 5

This also means that the nth term is a(n) = 17 - 3(n - 1)

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The image vertices of P′ are P′(-5, 1), S′(0, 1), R′(3, -2), and T′(-2, -4).

To determine the image vertices of P′ after rotating the preimage 90° counterclockwise, we can apply the following transformation rules for a 90° counterclockwise rotation about the origin:

(x, y) → (-y, x)

Using these rules, we can find the image vertices of the polygon:

P (1, 5) → P′ (-5, 1)

S (1, 0) → S′ (0, 1)

R (-2, -3) → R′ (3, -2)

T (-4, 2) → T′ (-2, -4)

Therefore, the image vertices of P′ are P′(-5, 1), S′(0, 1), R′(3, -2), and T′(-2, -4).

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Answer:

The volume of the figure is 76ft³.

Step-by-step explanation:

Divide the figure into 2 parts to make it easier to solve for the volume.

V1 = 6ft(2ft)(3ft)

V1 = 36ft³

V2 = 4ft(2ft)(5ft)

V2 = 40ft³

V = V1 + V2

V = 36ft³ + 40ft³

V = 76ft³

The graph is interpreted below

(a) the parabola open downward

(b) the coordinates of the vertex is (1, 9)

(c) the intercept(s)

x-intercept(s): (-2, 0) and (4, 0)

y-intercept(s): (0, 8)

(d) the equation of axis of symmetry: x = 2

To find the intercepts of a parabola, you need to determine the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept).

Examining the graph shows that the points are

x-intercept(s) (Roots): (-2, 0) and (4, 0)

y-intercept(s): (0, 8)

Also, investing the graph shows that the graph opens upward and the vertex which is the maximum point is (1, 9)

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The room temperature after 8 hours after the process begins would be 10°C.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + c

Where:

Based on the information provided above, a linear equation that models the cooling machine is given by;

y = mx + c

y = -5x + 50

By substituting the given parameter (x = 8 hours), we have the following:

y = -5(8) + 50

y = -40 + 10

y = 10°C

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Answer:

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

We know the tangent is perpendicular to radius.

Find the slope of the radius using its endpoints:

Perpendicular lines have negative reciprocal slopes, therefore the slope of the tangent is:

The matching choice is 2.

The equation that represents the situation is a = b- 4.

Since the plant grows at a constant rate, we can assume that the height of the plant is increasing linearly with time.

Let's use the data from Week 1 and Week 2 to find the rate of growth for each plant:

For Plant A: Growth rate = (24 cm - 22 cm) / (2 weeks - 1 week) = 2 cm/week

For Plant B: Growth rate = (28 cm - 26 cm) / (2 weeks - 1 week) = 2 cm/week

Since the growth rate is constant, we can use the equation of a line to model the height of each plant over time:

For Plant A: a = 2t + b, where t is the time in weeks and b is the initial height of the plant.

For Plant B: b = 2t + c, where c is the initial height of Plant B.

We can find the values of b and c by substituting the data from Week 1 into these equations:

For Plant A: 22 = 2(1) + b, so b = 20.

For Plant B: 26 = 2(1) + c, so c = 24.

Now we can substitute these values into the equations for Plant A and Plant B:

Plant A: a = 2t + 20

Plant B: b = 2t + 24

To find an equation that gives Plant A's height in terms of Plant B's height, we can solve the equation for t in terms of b:

b = 2t + 24

2t = b - 24

t = (b - 24) / 2

Then we can substitute this expression for t into the equation for Plant A:

a = 2t + 20

a = 2[(b - 24) / 2] + 20

a = b - 24 + 20

a = b - 4

So the equation we were looking for is:

a = b - 4

Therefore, to find Plant A's height in centimeters given Plant B's height in centimeters, we simply subtract 4 from Plant B's height.

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The salary plus benefits and the cost of living index for the two cities indicates;

1) Phoenix

2) San Antonio; $25,262.37, Phoenix; $21,901.35

3) San Antonio, because of the higher adjusted salary

The cost of living index is a measure of how expensive it is to reside in a city, compared with the national average costs.

1) The cost of living index for the cities indicates that Phoenix, with a cost of living index of 96.2 is more expensive to live in than San Antonio, Texas with a cost of living index of 85.3

2) The job offers can be evaluated by converting the salary from one city to another as follows;

(Salary in City A) × (Cost of living index for City B)/(Cost of living index for City A)

Therefore, we get;

San Antonio: ($17,000 + $5,400)×96.2/85.3 ≈ $25,262.37

Phoenix: ($23,500 + $1,200)*85.3/96.2 ≈ $21,901.35

3) The job offer that is better, based on the conversion of the job offers into adjusted salary for the two Cities of $25,262.37 vs $21,901.35, indicates that the San Antonio job offer is better

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The numerical value of x in the inscribed angle and intercepted arc is 4.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

From the image:

Inscribed angle XZY = 14x + 4

Intercepted arc XY = 30x

Value of x = ?

Plug these values into the above formula and solve for x.

Inscribed angle = 1/2 × intercepted arc.

14x + 4 = 1/2 × 30x

14x + 4 = 15x

15x = 14x + 4

15x - 14x = 4

x = 4

Therefore, x has a value of 4.

Option D) 4 is the correct answer.

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Answer:

Step-by-step explanation:

Sure! Here's an example of using tomato sauce and spaghetti as a ratio in a math problem:

Problem: Jane is making spaghetti and tomato sauce for dinner. She wants to mix the tomato sauce and spaghetti in a ratio of 1:4. If she uses 2 cups of tomato sauce, how many cups of spaghetti should she use?

Solution:

To solve this problem, we can set up a proportion using the given ratio:

Tomato sauce : Spaghetti = 1 : 4

Let's assign the variable 'x' to represent the number of cups of spaghetti.

Therefore, the proportion becomes: 1/4 = 2/x

To solve for 'x', we can cross-multiply and solve for 'x':

1 * x = 4 * 2

x = 8

Therefore, Jane should use 8 cups of spaghetti to maintain a ratio of 1:4 with 2 cups of tomato sauce.

You can include these calculations and steps to show how you arrived at the answer.

Step-by-step explanation: Here's an example of using tomato sauce and spaghetti as a ratio in a math problem:

Problem: Jannay has tomato sauce and spaghetti in a ratio of 1:4. If she uses 2 cups of tomato sauce, how many cups of spaghetti should she use?

Solution:

To solve this problem,

Tomato sauce : Spaghetti = 1 : 4

Assign the variable 'x' to represent the number of cups of spaghetti.

The proportion becomes: 1/4 = 2/x

To solve for 'x',  cross-multiply and solve for 'x':

1 * x = 4 * 2

x = 8

Therefore, Jannay should use 8 cups of spaghetti to maintain a ratio of 1:4 with 2 cups of tomato sauce.

Suppose,

[tex]y=(x-3)^2-47 ----------------(1)\\y=17 -------------------------(2)[/tex]

Put [tex]y=17[/tex] in place of x in the first equation:

[tex]17=(x-3)^2-47[/tex]

To both sides, add [tex]47[/tex]:

[tex]64=(x-3)^2[/tex]

Add the square roots of both sides together:

[tex]8=x-3[/tex] or [tex]-8=x-3[/tex]

To each sides, add [tex]3[/tex]:

[tex]x=11 or -5[/tex]

As a result, the following are the equations' solutions:[tex]x=-5, y=17 or x=11, y=17[/tex].

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Answer: 3:4

Step-by-step explanation: just divide it by 5

4x5=20

3*5=15

if the ratio was 6:15 then you would dive it by 2 because 2 is equivalent to 6 and 7.

Hope this helps :)

Answer:

6in

Step-by-step explanation:

angle C = 55 <-- 180-70-55=55
meaning the triangle is an isosceles so two sides would be the same.

A triangle with side lengths of 9, 40, and 41 is a valid right triangle.

The set of measurements 9, 40, and 41 could be the lengths of the sides of a right triangle. This is because these measurements satisfy the Pythagorean theorem,

which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, 9² + 40² = 41²,

which confirms that this set of measurements is valid for a right triangle. Additionally, 9 and 40 are both smaller than 41, which is consistent with the fact that the hypotenuse of a right triangle is always the longest side.

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Answer: let me explain!

Step-by-step explanation: So this might seem really confusing at first, but it’s actually suuuuuuper easy :P

So, if you think about it, every five seconds the amount of peanuts…well I don’t know how to explain it but let me show you this with numbers.
(4x2)x2x2x2 and so on. The number multiplies by two every five seconds from there. So figure out how many groups of five second there are in 100 seconds by dividing!

It’s 20!

So since 5 happens 20 times, and every time 5 happens, 2 happens, 2 also happens 20 times! (If that makes any sense)

So that answer would be 8x*twenty twos*

Which is…*drumroll please*

8,388,608
I even double checked!

And for the briefcase skeleton thing

She needs to try it 25 times because there are 5 books and each book can be switched with the other 4 times if the first one doesn’t work, therefore you need to multiply and the equation would be 5^2, or in other words, 25.
Don’t forget ur units!

Hope this helps :P

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where V is the volume and r is the radius.

Given that the sphere and the cylinder have the same radius and height, we can assume that the radius of both shapes is the same.

Let's denote the radius as 'r'. The volume of the cylinder is given as 50 ft³.

The volume of the cylinder is calculated using the formula V = πr²h, where h is the height.

Since the height of the cylinder is not given, we cannot determine the exact value of 'r' or calculate the volume of the sphere.

However, if we assume that the height of the cylinder is equal to its diameter (which is twice the radius), we can proceed with the calculation.

If the height of the cylinder is equal to 2r, the volume of the cylinder becomes:

50 = πr²(2r)

50 = 2πr³

Dividing both sides of the equation by 2π gives:

25 = πr³

Now, we can substitute this value of πr³ into the volume formula of the sphere:

V = (4/3)πr³

V = (4/3)(25)

V = 100/3

Therefore, if the height of the cylinder is equal to its diameter, the volume of the sphere would be 100/3 ft³. However, please note that this assumption might not necessarily be valid without further information about the relationship between the height and radius of the cylinder.

Answer: The Volume of the sphere is 100/3 cubic feet.

Step-by-step explanation:

It is clearly seen in the given figures, the height is two times the radius. Therefore, h is equal 2r.

We know that,

The volume of the sphere is π x r^2 x h. (r is radius and h is height)

So putting the values of height, h in the above formula, we get:

πxr^2x2r

= 2 * π * r^3.

= 2 * π * r^3 = 50

= π * r^3 = 25

We get, r^3 = 7.957747155

Therefore, r = 1.996472712

Now that we have the measure of the radius, we can find the volume of the sphere!

(4 / 3) * π * r^3

(4/3) * π * (1.996472712)^3

= (4/3) * π * 7.957747155

= (4/3) * π * (25 / π)

= (4/3) * 25

= 100/3 cubic feet.

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An estimated 3,042 people in attendance would prefer relish on a hot dog.

Given that;

For a recent football game of 8,450 in attendance,

And, 150 people were asked for they prefer on a hot dog.

Now, For number of people in attendance would prefer relish on a hot dog, we can use the proportion of people in the sample who preferred relish:

By proportion we get;

x = (54/150) x 8450

x = 54 * 56 1/3

x = 3042

Therefore, An estimated 3,042 people in attendance would prefer relish on a hot dog.

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The percent of sales tax on the food total is 6%.

We must first determine the 20% of the total amount of food that is the tip:

Tip = 0.2 * $32.50 = $6.50

The total cost of the meal before tax is:

Total cost before tax = food total + tip = $32.50 + $6.50 = $39.00

The cost of the lunch, including the sales tax of $1.95, is

Sales tax and the total cost before taxes = $39.00 + $1.95 =  $40.95.

The final price of the lunch, including tax and tip, is thus $40.95. You can compute the percentage of sales tax on the entire amount of food as follows:

Percent of sales tax on food = (sales tax / food total) * 100%

= ($1.95 / $32.50) * 100%

= 6%

So, the percent of sales tax on the food total is 6%.

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The coordinates in the solution to the systems of inequalities is (-2, 1)

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

y > -x - 4

y > 2x - 6

By imagining where the shading should be, we have

From the imagined graph, we have solution to the system to be the shaded region

The coordinates in the solution to the systems of inequalities is (-2, 1)

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Answer:

32) (x+12)(2)+(y)(2)=25

Step-by-step explanation:

(x-a)(2)+(y-b)(2)=r(2)

(x-(-12)(2)+(y-0)(2)=5(2)

(x+12)+(y)=25

Answer(s):

(32) - [tex](x +12)^2 + y ^2 = 25[/tex]

(33) - [tex](x +2)^2 + (y+6 )^2 = 36[/tex]

(34) - [tex](x -8)^2 + (y+1 )^2 = 100[/tex]

(35) - [tex](x -13)^2 + (y+12 )^2 = 1[/tex]

Step-by-step explanation:

To solve these types of problems, it is only a matter of memorizing the equation of a circle and plugging in values appropriately.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Equation of a Circle:}}\\\\(x - h)^2 + (y - k)^2 = r^2\\\text{Where} \ r \ \text{is the radius of the circle and} \ (h,k) \ \text{is the center of the circle.}\end{array}\right}[/tex]

For the following problems, find the equation of a circle with the given information.

(32) - [tex]\text{Center at} \ (-12,0) \ \text{and a radius,} \ r=5[/tex]

(33) - [tex]\text{Center at} \ (-2,-6) \ \text{and a radius,} \ r=6[/tex]

(34) - [tex]\text{Center at} \ (8,-1) \ \text{and a radius,} \ r=10[/tex]

(35) - [tex]\text{Center at} \ (13,-12) \ \text{and a radius,} \ r=1[/tex]

Question #32:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - (-12))^2 + (y - 0)^2 = (5)^2\\\\\Longrightarrow \boxed{ (x +12)^2 + y ^2 = 25}[/tex]

Question #33:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - (-2))^2 + (y - (-6))^2 = (6)^2\\\\\Longrightarrow \boxed{ (x +2)^2 + (y+6 )^2 = 36}[/tex]

Question #34:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - 8)^2 + (y - (-1))^2 = (10)^2\\\\\Longrightarrow \boxed{ (x -8)^2 + (y+1 )^2 = 100}[/tex]

Question #35:

[tex](x - h)^2 + (y - k)^2 = r^2\\\\\Longrightarrow (x - 13)^2 + (y - (-12))^2 = (1)^2\\\\\Longrightarrow \boxed{ (x -13)^2 + (y+12 )^2 = 1}[/tex]

Thus, all problems have been solved.