The roof shingles are arranged so that each row has 7 fewer shingles than the row below it. If the first row has 75 shingles, how many shingles are in the 6th row?

Answer:

The time taken for $200 invested at 7% annual interest compounded yearly:

[tex]\boxed{\mathrm{10.245 \; {years}}}[/tex]

which roughly works out to
[tex]\boxed{\mathrm{10\;years\;and\;3\;months}}[/tex]

Step-by-step explanation:

The formula for the accrued value of an amount P deposited at i% interest for a time period of t years compounded annually is given by the formula

[tex]\boxed{A = P(1 + r)^t\;\cdots[1]}[/tex]

Here A is the accrued value, P the principal and r = i/100

If you are given A, P and r we can compute t by taking the logarithms on both sides

In Equation [1] we get
[tex]\dfrac{A}{P} = (1 + r)^t \;\cdots[2][/tex]

Taking logs on both sides,
[tex]\log A - \log P = \log((1+r)^t)\\\\\log x^a = a \log x\\\\\\[/tex]

Therefore the right side becomes
[tex]t \log (1+r)[/tex]

Replacing right side of equation [2] with this expression yields

[tex]\log A - \log P = t \log (1+r)\\\\\textrm{Or, }\\\\t =\dfrac{ \log A - \log P}{\log (1 + r)}\\\\[/tex]

This is the general equation for determining how long it will take for the principal P to reach the value A if compounded annually at rate r(in decimal)

Since we are interested in seeing our principal P=200 double to A = 400 at r = 7% = 7/100 = 0.07
and
1 + r = 1 + 0.07 = 1.07

[tex]t = \dfrac{\log 400 - \log 200}{\log 1.07}\\\\[/tex]

[tex]t = \boxed{10.245 \; \textrm{years}}[/tex]

or approximately

[tex]\boxed{\mathrm{10\;years\;and\;3\;months}}[/tex]

Answer:

10.24 years (approx. 10 years 3 months)

Step-by-step explanation:

Most fixed annuities pay annual compound interest.

[tex]\boxed{\begin{minipage}{7 cm}\underline{Annual Compound Interest Formula}\\\\$ A=P\left(1+r\right)^{t}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

Given:

To calculate after how many years the value of the annuity will double, substitute the given values into the formula and solve for t:

[tex]\implies 400=200(1+0.07)^t[/tex]

[tex]\implies 2=(1.07)^t[/tex]

[tex]\implies \ln 2=\ln (1.07)^t[/tex]

[tex]\implies \ln 2=t \ln 1.07[/tex]

[tex]\implies t=\dfrac{\ln 2}{\ln 1.07}[/tex]

[tex]\implies t=10.2447683...\rm years[/tex]

Therefore, the value of the annuity will double after 10.24 years (approximately 10 years 3 months).

The proportion of students that had a grade at least 5 points greater than the 25th percentile is given as follows:

0.7054 = 70.54%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation for this problem are given as follows:

[tex]\mu = 78, \sigma = 36[/tex]

The 25th percentile is X when Z = -0.675, hence:

-0.675 = (X - 78)/36

X - 78 = -0.675 x 36

X = 53.7

The proportion of students with a grade of at least 58.7 is one subtracted by the p-value of Z when X = 58.7, hence:

Z = (58.7 - 78)/36

Z = -0.54

Z = -0.54 has a p-value of 0.2946.

1 - 0.2946 = 0.7054 = 70.54%.

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The sum of the first seven terms is given by -2186

Any progression in which the ratio of adjacent terms is fixed is a Geometric Progression. The general form of representation of a geometric sequence is a, ar, ar², ar³, ...

The geometric sequence is given here by -2,-6,-18...

We can observe that the common ratio is r=3

thus the sequence of upto 7 terms is given by

-2,-6,-18,-54,-162,-486.-1458

Thus the sum of all seven terms is given by

S=-2-6-18-54-162-486-1458

=-2186

Hence, The sum of the first seven terms is given by -2186

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Answer:

-1

Step-by-step explanation:

Add 2 to both sides of the equation: -7 + 2 = 5x - 2 + 2. This gives you -5 = 5x.

Divide both sides of the equation by 5: -5/5 = 5x/5. This gives you -1 = x.

So the solution to the equation is x = -1.

Alternatively, you can also use inverse operations to solve the equation by isolating x to one side.

Add 2 to both sides of the equation: -7 + 2 = 5x - 2 + 2. This gives you -5 = 5x

Divide both sides of the equation by 5: -5/5 = 5x/5. This gives you -1 = x

So the solution to the equation is x = -1.

The velocity after 3 second is -118 ft/sec and the velocity after falling 108 feet is -87.6 ft/sec.

(a)

For the given position function

[tex]s(t)=-16t^2+v_0.t+s_0[/tex]

The ball is dropped from top of the building of height 220 ft, therefore

[tex]s_0=220ft[/tex]

Also, the initial velocity of the ball is −22 feet per second, thus.

[tex]v_0=22\frac{ft}{s}[/tex]

We will differentiate the position function to find the velocity function and then calculate velocity at (a) t=3sec and (b) after failing 108 ft.

Differentiate Position function to acquire the velocity function v(t).

[tex]s(t)=-16t^2+v_0.t+s_0\\\\s'(t)=-32t+v_0[/tex]

Now substitute t=3s,

v(3) = -32(3) + (-22)

v(3)= -118 ft/sec

b)

To find the velocity of ball after falling 108 ft, first we need to find the time at which ball is dropped 108 ft. and then find velocity at that time using velocity function.

let s(t)=108 ft, then using position function, we can write

[tex]s(t)=-16.t^2+v_0t+s_0\\\\108=-16t^2+(-22)t+220\\\\-16t^2-22t+112=0[/tex]

After solving the quadratic equation, we get

t= -3.42 and 2.05

Since time always positive,

therefore t=2.05 sec.

v (2.05) = -32(2.05)-22

v (2.05) = -87.6 ft/sec

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In trigonometry it is important that the right triangles (reference triangles) have a consistent orientation. the reference triangles always have a leg perpendicular to the y-axis , the value of the hypotenuse is positive , and is the angle adjacent to the y-axis and hypotenuse.

A right angle triangle has six trigonometric ratios: Sin, Cos, Tan, Cosec, Sec, and Cot. They represent, successively, Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent.

So, when we graph a right angle triangle, we draw a side of triangle perpendicular to y-axis and another side will be on y-axis. The remaining side left would be hypotenuse that will always be positive and is adjacent to the side perpendicular to y-axis and the side on the y-axis.

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There may be a correlation, but there is no direct link. This is because it is more likely that you will get shoes in the color you want.

It's likely that the preferred color and the color of the shoes are related. A person's favorite color may or may not match the color of their shoes, so there is no direct correlation. Particularly, the color of the shoes may have influenced the preferred color.

There may be a correlation, but there is no direct link.

Correlation is a method for figuring out how two variables relate to each other. You learned that putting two variables on a "scatter plot" can help you figure out if they are usually connected. Despite the fact that there are other measures of association for variables measured at the ordinal or higher level of measurement, correlation is the strategy that is utilized the most frequently.

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The fractions with different numerators and denominators but the same value are said to be equivalent fractions.

For instance, since 2/4 and 3/6 both equal the 1/2, they are equivalent fractions. An element of a whole is a fraction. The same amount of the whole is represented by equivalent fractions.

A rule that says the result of multiplying the numerator and denominator of a fraction by the same nonzero number is a fraction equal to the original fraction.

Equivalent fractions are those that, despite their visual differences, represent the same value. For instance, if you have a cake and cut it into two equal pieces, you will have consumed half of the cake.

In this case, it should be noted 1/2 is the same as 3/6.

Note that an overview was given as your information is incomplete.

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The p-value is the chance of finding a t-value with an absolute value at least as great as the one we actually observed in the sample data assuming the null hypothesis is actually true. The t-value is a way to measure the difference between the population means.

We reject the null hypothesis of the test if the p-value is less than a predetermined threshold (for example, 0.05). We are primarily concerned with the p-value for each form of t-test, and we merely employ the t-value as a first step in that process.

You'll see that we merely used the t-value as a first step before determining the p-value. The genuine value in which we were most interested was the p-value, but first, we had to determine the t-value.

Therefore the p-value is the likelihood that, if the null hypothesis is correct, each test will result in a t-value with an absolute value at least as great as the one we actually observed in the sample data. The t-value is a way to measure the difference between population means.

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In a case whereby a right triangle, sin(7x+5)=cos(8x-5),  the smaller of the triangles two acute angles is 43°.

The concept that will be used is  right triangle. An acute angle is one that is smaller than 90 degrees in length. The correct angle is larger than this angle (which is equal to 90 degrees).

We can see  that sin(7x+5)=cos(8x-5)

An it implies that 7x+5 -8x-5  =90°

15x=90

x= 6

Then 7(6)+5= 47°     8(6)-5 = 43°

Hence the smaller of the triangles two acute angle is 43°

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It would take 17 days to build a similar house with 25 workers.

Inverse proportion is a type of proportionality relationship. If two quantities are inversely proportional then as one quantity increases, the other decreases.  In inverse proportion, the product of the given two quantities is equal to a constant value

One house was built in 85 days by 5 workers;

The relationship will be;

d ∝ 1/n

where

n = number of workers

d = number of days

d = k/n

when d = 5

k = 85

k = 5 x 85

k = 425

when n = 25

d = 425/25

d = 17

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a) The function is P = 500e^0.4t

b) The count after 10 hours is 27299

c) it would reach 80000 in 13 hours

We know that at this time we are dealing with the population of the bacteria and we have to invoke the exponential growth ideology and we would have;

P = Poe^rt

P = population at time t

Po = Initial population

r = Population growth rate
t = Time taken

Then we have;

P = 500e^0.4t

After ten hours;

P= 500e^0.4 * 10

P = 27299

The count would reach 80000 when;

80000 = 500e^0.4 t

80000/500 = e^0.4 t

160 = e^0.4 t

ln160 = e^0.4 t

ln160 = 0.4t

t = ln160/0.4

t = 13 hours

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166158  is the break-even point for the manual process

An expression is combination of variables, numbers and operators.

Given that The manual process will have a fixed cost of $36,234 and a variable cost of $2.14 per bag.

Let the number of bags will be x

36234+2.14x..(1)

The mechanized process would have a fixed cost of $84,420 and a variable cost of $1.85 per bag.

84420+1.85x..(2)

From equation 1 and 2

36234+2.14x=84420+1.85x

2.14x-1.85x=84420-36234

0.29x=48186

Divide both sides by 0.29

x=166158

Hence, 166158  is the break-even point for the manual process

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The value of the variable , x is 10. 33

It is important to note that in mathematics, complementary angles are defined as angles that sum up to 90 degrees.

Some properties of complementary angles are;

From the information given, we have the angles as;

Equate the angles to 90 degrees

8x - 9 + x + 6 = 90

collect like terms

8x + x = 90 +3

add the like terms, we get;

9x = 93

Make 'x' the subject of formula by dividing both sides by 9

x = 93/9

x = 10. 33

Hence, the value is 10.33

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Answer:

Step-by-step explanation:

The slope's simplest form is 1/2.

The number of years that it will it have about 100 students is 11 years.

Final population = Initial population * (1-0.04)^years

Where 0.04 is the percentage decrease (4%) converted to decimal form, and "years" is the number of years in the future.

We can use this formula to find the number of years it will take for the senior class to have about 100 students:

Final population = 100

Initial population = 160

100 = 160 * (1-0.04)^years

We can solve for "years" by taking the natural logarithm of both sides:

ln(100) = ln(160 * (1-0.04)^years)

ln(100) = ln(160) + ln((1-0.04)^years)

and then we can solve for the power of (1-0.04)

ln(100) - ln(160) = ln((1-0.04)^years)

now we can get the value of years

years = (ln(100) - ln(160)) / ln(1-0.04)

which is around 10.72 , so it would take about 11 years for the senior class to have about 100 students.

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The expanded value of the given equation is a²+2bx = b² + x ²+ h² .

Multiplication distributes over addition, so an expansion of a product of sums represents it as a sum of products.

Expansion is a scale-increasing affine transformation, sometimes known as enlargement or dilatation. It is also frequently referred to as an enlargement and is the polar opposite of a geometric contraction. An expansion and a translation are equivalent to a central dilation.

We must multiply out the brackets in order to expand and simplify an expression, and we must then collect like words in order to simplify the resulting expression.

Given : a^2= (b-x)^2 +h^2

          a² = (b-x)² + h²

we know that (c-d)² = c²+d² + 2cd

So,    a² = b² + x² -2bx + h²

         a²+2bx = b² + x ²+ h²

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Answer:

3% = 56

100% = ?

Rule of three:

56 x 100

—————

3

Answer = 1,866.667 ≈ 1.867

Step-by-step explanation:

dont mind the answer below have a great day ______

                                                                                            \|/            

A coordinate plane is a two-dimensional graph that is used to locate points in a plane using an x-axis and a y-axis.

The slope of the line is (4-(-4))/(7-1) = 8/6 = 4/3.

The y-intercept is -4.

So the equation of the line is y = (4/3)x - 4.

We know that the area of triangle BCD is 15 square units. To find the possible coordinates of point D, we can use the formula for the area of a triangle (A = (1/2)bh). Since we know the area and the base, we can find the height of the triangle and use that to find the possible coordinates of point D.

The equation of the line is y = (4/3)x - 4. We can use this equation to find the y-coordinate of point D.

y = (4/3)x - 4

15 = (1/2)(x)((4/3)x - 4)

By solving the equation above, we can find the possible x coordinate of point D, which are (-2, -4/3) and (12, 4). Therefore, the possible coordinates of point D are (-2, -4/3) and (12, 4).

Note: you can use the same formula to find the y coordinate if you were given the x coordinate of D, but it gives the same solution.

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While the new helmets decrease the probability of injury resulting from a scooter accident, but they also incentivize individuals to ride more recklessly, which could increase the number of scooter collisions and therefore injuries to renters.

The reason for the above statement is based on the concept of risk compensation. When the perceived risk of an activity is reduced, individuals may engage in more risky behavior. For example, if individuals believe that wearing a helmet makes them safer while riding a scooter, they may feel less concerned about the potential consequences of riding recklessly, leading to an increase in the number of accidents. On the other hand, the helmet does decrease the probability of injury in the event of a crash, but an increase in the number of crashes could lead to an overall increase in the number of injuries, offsetting the protective effect of the helmets. Therefore, the net impact of the new helmets on the number of scooter injuries is unclear, and further study is needed to determine their overall effect.

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Answer:

[tex](0,-2)[/tex]

Step-by-step explanation:

Let the coordinates of the desired point be [tex](a, -a^2-2)[/tex]. Let the distance of this point from [tex](0,1)[/tex] be [tex]f(a)[/tex].

By the distance formula, [tex]f(a)=\sqrt{a^2+(a^2+3)^2}[/tex].

Clearly, [tex]f(a)[/tex] is minimized when the radicand is minimized.

Clearly, the radicand is always increasing, so the minimum is achieved at the left endpoint of the maximal domain of [tex]f(a)[/tex], which is [tex]a=0[/tex].

So, the coordinates are [tex](0,-2)[/tex].

The point on the curve y = -x^2 - 2 that is closest to the point (0, 1) will be (0, -2)

From the information, we want to know the point on the curve y = -x^2 - 2 that is closest to the point (0, 1).

Let the coordinates of the desired point be illustrated as (a - a² - 2).

The distance of the point from (0, 1) is depicted as f(a).

When the radicand is minimized, f(a) is minimized.

In this case, the minimum will be achieved at the left endpoint of the maximal domain of f(a) which is a = 0.

Therefore, the coordinates will be (0, -2).

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Answer: The total bill including tax and gratitude is $35.20

And each one of the 5 friends paid $10 for the bill.

So the total amount paid is $10*5 = $50

If each person paid $10 and the total bill was $35.20

then the change will be $50-$35.20 = $14.80

And since they split the bill equally then each one of them will get back $14.80/5 = $2.96

So each person will get back $2.96

Step-by-step explanation:

Using the empirical rule, 95% of vehicles travel between 60 miles per hour and 80 miles per hour.

Generally, Given that

To find the percent of vehicles whose speeds are between 60 miles per hour and 80 miles per hour

60 = 70-(2*5) ,

this shows that the 58 value is 2 standard deviations below the mean

and

80 = 70 +( 2*5)

This shows that the 74 value is 2 standard deviations above the mean

Using the empirical rule, we know that 95% of data fall within 2 standard deviations from the mean.

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If a particular sample has 36 participants and the other sample has 40 participants, the total degrees of freedom for the study is

In Mathematics, the degrees of freedom in a sample are typically used to correct a possible bias that exists between the alternative and null hypotheses.

Mathematically, total degrees of freedom for the study can be calculated by using this mathematical expression:

v = n - k

Where:

Note: k is equal to 2 in this scenario.

Total degrees of freedom, v = n₁ + n₂ - k

Total degrees of freedom, v = 36 + 40 - 2

Total degrees of freedom, v = 74.

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Using the Time-Distance Formula, it will take Adbi 2 hours.

We can use the time-distance formula to find how long it will take for Abdi to catch up with Bashir. The time-distance formula is distance = speed x time.

Let t be the time in hours that it takes for Abdi to catch up with Bashir.

We know that the distance Abdi has covered is 4 miles more than Bashir, and the relative speed between the two runners is 8mph - 6mph = 2mph.

So we can write the equation:

4 = (8 - 6) * t

Solving for t, we get t = 4/2 = 2 hours.

So it will take Abdi 2 hours to catch up with Bashir.

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The total area of the garden is given as follows:

32.1 m².

The total area of the garden is obtained applying the proportions in the context of this problem.

The area of native plants is given as follows:

107/4 = 26.75 m².

This area of 26.75 m² represents a fraction of 5/6 of the total area g of the garden, hence the total area of the garden is obtained applying the proportion as follows:

5g/6 = 26.75.

g = 26.75 x 6/5

g = 32.1 m².

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The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population.

The population parameter is the true value of the statistic for the entire population. The probability that the statistic is obtained is the probability that it will be obtained in a single sample from the population. The variance of the values is the amount of variation among the values of the statistic in the sample. The formula for calculating the variance is the sum of the squared deviations from the mean, divided by the number of values in the sample. This can be written mathematically as: Variance = Σ (x - X)2/ n, where x is each value in the sample, X is the mean value, and n is the number of values in the sample.

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The equation that can be used to find how many hours, x, the second part of the trip will take is:

distance = rate x time

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The boxes are split in halves, and there is 4 boxes. The answer is 4 / 1/2.

What is division?

Division is a mathematical operation that involves dividing one number (the dividend) by another (the divisor). The result of this operation is called the quotient. Division is the inverse of multiplication, meaning that the product of a number divided by itself is equal to one. Division can also be used to find the number of times one number is contained in another. Division can be expressed in a variety of ways, such as by using fractions, decimals, or by using a long division algorithm.

When you look at the diagram, The boxes are split in halves, and there is 4 boxes. The answer is 4 / 1/2.

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The fraction that voted for the gym is calculated as: 5/28.

To find out what fraction voted for the gym, you have to subtract the fractions that voted for the restaurant and the library from 1, which represents the total number of people surveyed.

You can do this by adding the fractions that voted for the restaurant and the library and then subtracting the sum from 1.

(1/4) + (4/7) = (7/28) + (16/28) = (23/28)

so 1 - (23/28) = (5/28)

Therefore, 5/28 of the people surveyed voted for the gym.

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