The sides of a triangle are 30, 64, and 90. Use the Pythagorean Theorem to determine if the triangle is right, acute, or obtuse.

By mathematical induction, the given formula holds for all natural numbers n:

S(n) = 36 + 324 + 900 + ... + (12n - 6)² = 12n(4n² - 1)

To demonstrate our point, mathematical induction can be employed.

We want to prove that the given formula holds for all natural numbers n:

S(n) = 36 + 324 + 900 + ... + (12n - 6)² = 12n(4n² - 1)

Using mathematical induction:

Step 1: Base case (n=1)

S(1) = 6² = 36

12 * 1 * (4 * 1² - 1) = 36

Since S(1) holds true, the base case is proven.

Step 2: Inductive step (Assume it's true for n=k, and show it's true for n=k+1)

Assumption: S(k) = 36 + 324 + 900 + ... + (12k - 6)² = 12k(4k² - 1)

We follow the exhaustive process to prove:

= 12(4k³ + 12k² + 11k + 3)

Since both expressions for S(k+1) are equal, the inductive step is proven.

Thus, by mathematical induction, the given formula holds for all natural numbers n:

S(n) = 36 + 324 + 900 + ... + (12n - 6)² = 12n(4n² - 1)

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In a case whreby the electrical potential between two electrons is given by a formula has the form 1/r + 1/ 1-r the simpification of the expression is 1/r(1-r).

Looking at the expression we can see that it is adion of two expression which implies that we will need to perform an addition operation then later simplified the result, this is been done below

We can find the LCM as r(1-r)

(1-r +r)/ r(1-r)

=1/r(1-r)

Therfore, after performing the addition operation as well as the simplification, then the the simpification of the expression is 1/r(1-r).

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complete question;

The electrical potential between two electrons is given by a formula has the form 1/r + 1/ 1-r simplify the expression.

Answer:

6

Step-by-step explanation:

this is if you meant: Jasmine sold 5 cherry pies and 12 lemonade how much more did Jasmine sell of lemonade than cherry pies

The diameter of the circle is 16.04 cm.

The circumference is 50.39 cm

A = πr²

where A is the area of the circle and r is its radius.

square both sides

r = √(A/π) = √(201.06/π) = 8.02 cm

The diameter of the circle is twice the radius, so we can find it by multiplying the radius by 2:

d = 2r = 2(8.02) = 16.04 cm

Therefore, the diameter of the circle is 16.04 cm.

To find the circumference of the circle, we can use the formula:

C = 2πr

where C is the circumference and r is the radius. We already know the value of the radius, so we can substitute it into the formula:

C = 2π(8.02) = 50.39 cm

Therefore, the circumference of the circle is 50.39 cm.

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A square of a binomial is a polynomial expression that results from multiplying a binomial by itself. The result is a trinomial expression with three terms.

Answer:

If y and x have a proportional relationship, then we can write an equation of the form y = kx, where k is a constant of proportionality.

To find the value of k, we can use the given values of y and x:

y = kx

24 = k(12)

k = 2

Now that we know k, we can use the equation to find the value of y when x=16:

y = kx

y = 2(16)

y = 32

Therefore, when x=16, y=32.

The ratio that could also be included in this table is (c) 18 : 30

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

Games won   Games played

6                    10

24                  40

36                60

42                70

From the above, we have the ratio of the number of games won to the number of games played to be

Ratio = 6 : 10

Simplify

Ratio = 3 : 5

From the options, we have

(c) 18 : 30

Simplify

Ratio = 3 : 5

Hence, the ratio that could also be included in this table is (c) 18 : 30

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Answer:

The answer is

x= -2

y= -8

Step-by-step explanation:

x=y+6

substitute into equation 1

13(y+6)-6y=22

13y+78-6y=22

13y-6y=22-78

7y=-56

divide both sides by 7

7y/7= -56/7

y= -8

x=y+6

x= -8+6

x= -2

Answer:

First, you need to calculate the total cost of the loan. The monthly payment is $260, and there are 72 payments over 6 years. The total cost of the loan is 72 months * $260/month = $18,720

Next, you calculate the total cost of owning the car for 6 years. The insurance will cost $960 per year, so over 6 years it will cost $960/year * 6 years = $5,760

The registration will cost $240 per year, so over 6 years it will cost $240/year * 6 years = $1,440

The gas and maintenance will cost $175 per month, so over 6 years it will cost $175/month * 72 months = $12,600

Now you can add up all the costs to find the total cost of owning the car over the life of the loan:

Loan cost + insurance cost + registration cost + gas/maintenance cost

= $18,720 + $5,760 + $1,440 + $12,600

= $38,520

Therefore, the car will cost Brooke $38,520 over the life of the loan, including all additional costs of owning the car.

Answer:

$38,520

Step-by-step explanation:

To calculate the total cost of the car, we need to consider the following expenses:

Monthly payments:

$260/month x 72 months = $18,720

Insurance:

$960/year x 6 years = $5,760

Registration:

$240/year x 6 years = $1,440

Gas and maintenance:

$175/month x 12 months/year x 6 years = $12,600

Therefore, the total cost of the car over the life of the loan is:

$18,720 (monthly payments) + $5,760 (insurance) + $1,440 (registration) + $12,600 (gas and maintenance) = $38,520

So, while Brooke may think she's getting a good deal with zero down and low monthly payments, the total cost of the car over the life of the loan is quite high. It's important to consider all the additional costs associated with owning a car before making a decision to purchase one.

Answer: Based on the given figure above, we can conclude that the triangle is an isosceles triangle. By definition, an isosceles triangle is a triangle that has at least two equal sides. Since this is an isosceles triangle, 8x-10 =6x. Now we can solve for x. So,

8x-10 =6x

8x-6x = 10

2x =10

x= 5.

Therefore, the value of x in the figure is 5. Hope this is the answer that you are looking for.

Callie can make 5 necklaces.

So to convert from a value in inches to one in yards, we need to divide by 36. To convert a value in yards to inches, we need to multiply by 36.

Now, She has a pieces of leather cord that has a length of 4 yards.

Each necklace will have a length of 28 inches.

We have to determine the how many necklaces can Callie make?

We know that :

1 yard= 36 inches

So, Multiply by 36 into 4 yards

4 yards= 36x4= 144

Length of leather cord is 4 yards

144/4 = 5.14

Callie can make 5 necklaces.

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Answer:

A quadrilateral is a shape that has four straight sides and four angles.

A triangle is a shape that has three straight sides and three angles.

A square is a special type of quadrilateral that has four straight sides of equal length and four right angles.

The equation for the rectangle is adding the interior angles:

x + x - 5 + 71 + y + 21 = 360°

Solving that for y, we will get:

2x + y + 87 = 360

Remember that the sum of the interior angles of any figure with 4 sides must be 360°.

Then we can write an equation:

x + x - 5 + 71 + y + 21 = 360°

2x + y + 87 = 360

Solving it for y:

y = 273 - 2x

Now there is a clear problem in the diagram because the angle in the bottom left is clearly larger than 90°, but it says that the angle measures 71°, so there we have a problem.

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The value of the median number of eggs laid in a year is,

⇒ 231

We have to given that;

The data table shows the numbers of eggs laid by individual chickens in a year.

Now, We can arrange it into ascending order as;

⇒ 222, 229, 229, 231, 234, 235, 262

Since, There are 7 terms

Hence, Median is,

= (7 + 1)/2

= 4th term

= 231

Thus, The value of the median number of eggs laid in a year is,

⇒ 231

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Answer:

A. 229 is correct

answer D.231 is wrong

Answer:

The points selected on the graph are:

(0, 0), (2, 4), (3, 6), (4, 8), (6, 12), (4, 8), (8, 16), (5, 10)

Here is a table showing the coordinates of the selected points:

x      y

0      0

2      4

3      6

4      8

6      12

4      8

8      16

5      10

If owner of car paid $780 as annual premium, then the cost of the accident is $26540 more expensive than what car owner paid in premiums, correct option is (d).

The car owner paid a total of $780 per year for 7 years, which means that the total amount paid in premiums is:

⇒ Total premiums paid = $780 × 7 years = $5460,

When the owner caused an accident, the amount of damage claimed by other driver is $32000.

This means that cost of accident was : $32000 - $5460 = $26540 more expensive than what the "owner" had invested in his insurance policy.

Therefore, Option(d) is correct.

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The given question is incomplete, the complete question is

A car owner pays an annual premium of $780 for automobile insurance, including liability coverage of up to $100,000. The car owner pays this for seven years without needing to file a

single claim. Then the car owner causes an accident for which the other driver is claiming $32,000 in damages. How much more expensive were the costs of the accident than what the car owner paid in premiums?

(a) $37,460

(b) $100,000

(c) $5,460

(d) $26,540

The expression (7√98 + 3√242) divided by 6√50 simplifies to the fraction of 4/3.

To simplify the expression (7√98 + 3√242) ÷ 6√50, we need to rationalize the denominator. This means we need to simplify the expression in such a way that there are no radical signs in the denominator. To do this, we can use the fact that the product of a radical and its conjugate is a rational number.

First, let's simplify the terms under the radicals by factoring out the perfect squares:

√98 = √(49*2) = 7√2

√242 = √(121*2) = 11√2

Now the expression becomes:

(7√2 + 311√2) ÷ (6√(252))

= (7√2 + 33√2) ÷ (6*5√2)

= 40√2 ÷ 30√2

= 4/3

In conclusion, to simplify expressions involving radicals, we need to factor out perfect squares and use the property of the conjugate to rationalize the denominator.

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The value of x is 5.48 units

Let us assume that the left right triangle be ABC and the right triangle on the right side be ABD.

So, the larger right triangle would be DAC.

Consider thr outer right triangle DAC.

The length of the hypotenuse DC would be,

DC = CB + BD

DC  = 3 + 7

DC = 10 units

Also right triangle ABC and DAC are similar triangles.

We know that the sides of the smilar triangles are in proportion.

Using the definition of similar triangles for DAC and ABC,

DA/AB = AC/BC = DC/AC

Consider, AC/BC = DC/AC

x/3 = 10/x

this is the required proportion.

x² = 3 × 10

x² = 30

x = 5.48 units

Thus the value of x = 5.48 units

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Answer:  -5, 3

Step-by-step explanation:

Equation:

x² + 2x -15 = 0   The equation is already set =0   so you will factor.

To factor, find 2 numbers that multiply to the last term (-15) but add up to the middle term (+2).  Be careful with signs,  signs matter

+5 and -3 multiply to -15 but add up to the +2    so these are your factored numbers

put it in factored form

(x+5)(x-3)=0         set each parenthesis =0 to solve for x

x+5 =  0                    x-3=0

x= -5        and           x=3

The option that is a representation that shows y as a function of x is the graph in option 2. The answer is B.

A function is any relation or table of values which may be represented in a graph that has only one possible y value that is assigned to each x-value.

The first option is not a function because x-value 0 has two corresponding y-values, 4 and 9.

In the third option, we also have two y-values, 5 and -5, that is assigned to one x-value, 0. So, it is not a function.

The graph in option 2 represents y as a function of x, because no two y-values is assigned to the same x-value.

The answer is: B.

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The domain of the relation, given the points the coordinate plane has, is A.{−5, −3, 0, 2, 5}

The range of representable x-values in a function or relation when plotted on a coordinate plane are regarded as the domain. This values serve as inputs that can generate correspondig outputs within functions and relations. Displayed along the x-axis, domains strictly adhere to their input-based designation.

The x values from the graph are, - 5, - 3, 0, 2, 5 .

This means that the domain is {−5, −3, 0, 2, 5}.

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The chances of rolling between 20 and 40 aces (inclusive) in 180 rolls is approximately 0.9544 or 95.44%.

We can first standardize the values of 20 and 40 using the formula:

z = (x - μ) / σ

where x is the value we want to standardize (20 or 40), μ is the expected value (30), and σ is the standard deviation (5).

For x = 20, we get:

z = (20 - 30) / 5 = -2

For x = 40, we get:

z = (40 - 30) / 5 = 2

probability of getting a z-score between -2 and 2.

Using a standard normal distribution table or calculator, we find that the probability of getting a z-score between -2 and 2 is approximately 0.9544.

Therefore, the chances of rolling between 20 and 40 aces (inclusive) in 180 rolls is approximately 0.9544 or 95.44%.

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Based on the information, we can infer that the investment was made 11.53 years ago.

To calculate how many years ago was the investment mafe we have to can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = the final amount

P = the principal (initial amount invested)

r = the interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

Let's start by finding the initial principal (P) by working backwards from the final amount (A):

A = P(1 + r/n)^(nt)

31372.0 = P(1 + 0.022/1)^(14)(1 + 0.022+0.007/1)^(1(t-4))

Simplifying, we get:

31372.0 = P(1.093248)(1.029)^(t-4)

P = 27500 euros (initial investment)

Now we can solve for t:

31372.0 = 27500(1.093248)(1.029)^(t-4)

1.1412 = (1.029)^(t-4)

log(1.1412) = log(1.029)^(t-4)

t - 4 = log(1.1412)/log(1.029)

t = log(1.1412)/log(1.029) + 4

t = 11.53

Therefore, the investment was made about 11.53 years ago.

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Answer:

the corresponding sides are proportional

Answer:

Each small box has 24/17 beads.

Step-by-step explanation:

Let's call the number of beads in each small box "s" and the number of beads in each large box "l".

We know that there are 14 small boxes, so the total number of beads in the small boxes is 14s. We also know that there are 3 large boxes, so the total number of beads in the large boxes is 3l.

From the problem, we know that there are 4 more beads in each large box than in each small box, so we can write:

l = s + 4

We also know that 7/9 of the total number of beads were packed in the small boxes, so we can write:

14s + 3l = 7/9(total number of beads)

We can simplify this expression by substituting l = s + 4:

14s + 3(s + 4) = 7/9(total number of beads)

Expanding and simplifying:

17s + 12 = 7/9(total number of beads)

Now, we need one more equation to solve for s. We know that the total number of beads is equal to the sum of the beads in the small and large boxes:

total number of beads = 14s + 3(s + 4)

Simplifying:

total number of beads = 17s + 12

Now we have two equations that we can use to solve for s:

17s + 12 = 7/9(total number of beads)

total number of beads = 17s + 12

We don't know the total number of beads, but we do know that it is a multiple of both 14 and 3 (since there are 14 small boxes and 3 large boxes). The smallest multiple of both 14 and 3 is 42, so let's assume that the total number of beads is 42x, where x is some number.

Substituting 42x for "total number of beads" in the two equations:

17s + 12 = 7/9(42x)

42x = 17s + 12

Simplifying:

153s + 108 = 98x

42x = 17s + 12

Now we have two equations and two unknowns (s and x). We can solve for s in terms of x by isolating s in the second equation:

17s = 42x - 12

s = (42x - 12)/17

We can substitute this expression for s into the first equation:

17((42x - 12)/17) + 12 = 7/9(42x)

Simplifying:

42x - 12 + 12 = 14x

28x = 24

x = 24/28 = 0.857

Now we know that the total number of beads is:

42x = 42(0.857) = 36

And we know that each small box has:

s = (42x - 12)/17 = (36 - 12)/17 = 24/17

So each small box has 24/17 beads.

Answer:

|-34|, |-25|, |0.5|, |0.35| is the correct order.

The surface area of the rectangular prism in this problem is given as follows:

A. 426 m².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas. -> net method.

The dimensions for this problem are given as follows:

5 m, 12 m and 9 m.

Hence the surface area of the prism is given as follows:

S = 2(5 x 12 + 5 x 9 + 12 x 9)

S = 426 m².

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Using rules for logarithms we can see that the statement is false.

Here we have the equation:

logₐ(x^3 + y^3) = 3logₐ(x) + 3logₐ(y)

And we want to see if it is true.

First, remember two rules:

logₐ(x^n) = n*logₐ(x)

and:

logₐ(x) + logₐ(y) = logₐ(x*y)

Then, the right side of the statement can be rewritten as:

3logₐ(x) + 3logₐ(y)

3(logₐ(x) + logₐ(y)) = 3*logₐ(x*y) = logₐ((x*y)^3)

And that is different to what we got in the statement, so the statement is false.

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(a) The lesson time in 2007 is 11.1% more than the lesson time in 2006.

(b) The lesson lasted 40 minutes in 2005.

(c) James will take 180 seconds to swim 100 meters in a year's time.

(d)  Robert will also take 180 seconds to swim 100 meters in a year's time.

(e) James and Robert will swim equally which means that their speed will be same and they will be complete at same time.

(a) The increase in lesson time from 2006 to 2007 is:

Increase = 50 - 45 = 5 minutes

The percentage increase is:

Percentage increase = (5 / 45) x 100% = 11.1%

Therefore, the lesson time in 2007 is 11.1% more than the lesson time in 2006.

(b) Let x be the lesson time in 2005. The lesson time in 2007 is 25% more than it was in 2005, so:

1.25x = 50

Solving for x, we get:

x = 40

Therefore, the lesson lasted 40 minutes in 2005.

(c) James currently takes 240 seconds to swim 100 meters. If he improves his time by 25%, his new time will be:

New time = 240 - (0.25 x 240) = 180 seconds

Therefore, he will take 180 seconds to swim 100 meters in a year's time.

(d) Robert currently takes 240 seconds to swim 100 meters. If he increases his speed by 25%, his new time will be:

New time = 240 - (0.25 x 240) = 180 seconds

Therefore, he will also take 180 seconds to swim 100 meters in a year's time.

(e) Both James and Robert will take the same time to swim 100 meters in a year's time, so they will be equally fast.

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1. intrest earned per year $116.25

2. total after 20 years $3,825

116x20= $2325

$1500+ 2325=3825