Here is an equilateral triangle. The length of each side is units. A height is drawn. In an equilateral triangle, a line drawn from one corner to the center of the opposite side represents the height. 1. Find the exact height. 2. Find the area of the equilateral triangle. 3. (Challenge) Using for the length of each side in an equilateral triangle, express its area in terms of .

Answer:

  0.8

Step-by-step explanation:

The change of base formula for logarithms can be used to find this value exactly. (Your calculator can also tell you.)

  [tex]\log_b(a)=\dfrac{\log(a)}{\log(b)}[/tex]

__

Using this relation with the given numbers, we have ...

  [tex]\dfrac{\log(256)}{\log(1024)}=\dfrac{\log(2^8)}{\log(2^{10})}=\dfrac{8\cdot\log(2)}{10\cdot\log(2)}=\dfrac{8}{10}\\\\\boxed{\dfrac{\log(256)}{\log(1024)}=0.8}[/tex]

To find the length of the diagonal, we need to calculate the length of the base first and then use the value to calculate the diagonal since we are given the value of height of the triangle.

This theorem is used to find the missing side of a right angle triangle given that we have the length of two sides.

To calculate the base of the triangle, let's use Pythagorean theorem here

[tex]x^2 = y^2 + z^2\\x^2 = 2^2 + 3^2\\x^2 = 4 + 9 \\x^2 = 13\\x = \sqrt{13} \\x = 3.61[/tex]

Having the length of the base, let's consider the height of the triangle which is 6 and substitute the values.

[tex]d^2 = 6^2 + 3.61^2\\d^2 = 36 + 13.0321\\d^2 = 49.0321\\d = \sqrt{49.0321} \\d = 7.00[/tex]

From the calculations above, the values of the triangle are

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Step-by-step explanation:

p = 2r + pi×r

p = r(2 + pi)

r = p/(2 + pi)

Answer:

r = [tex]\frac{p}{2+\pi }[/tex]

Step-by-step explanation:

p = 2r + πr ← factor out r from each term

p = r(2 + π) ← divide both sides by (2 + π )

[tex]\frac{p}{2+\pi }[/tex] = r

Answer:

3,-8

Step-by-step explanation:

x: from -3 to 0   is  +3    add +3 more     3

y: from -2 to -5 is   -3     add -3 more    -8

Answer:

[tex]ax^5+ by^5=241[/tex]

Step-by-step explanation:

Given:

We can re-write the left sides of the given equations as follows:

[tex]ax^2+ by^2=(ax+by)(x+y)-xy(a+b)[/tex]

[tex]ax^3+ by^3=(ax^2+by^2)(x+y)-xy(ax+by)[/tex]

[tex]ax^4+ by^4=(ax^3+by^3)(x+y)-xy(ax^2+by^2)[/tex]

Therefore, following this pattern:

[tex]ax^5+ by^5=(ax^4+by^4)(x+y)-xy(ax^3+by^3)[/tex]

Use the given values and the expanded expressions to create 2 equations to help find the values of (x+y) and xy:

Equation 1

[tex]ax^3+ by^3=(ax^2+by^2)(x+y)-xy(ax+by)[/tex]

[tex]\implies 25=11(x+y)-xy(1)[/tex]

[tex]\implies 25=11(x+y)-xy[/tex]

Equation 2

[tex]ax^4+ by^4=(ax^3+by^3)(x+y)-xy(ax^2+by^2)[/tex]

[tex]\implies 83=25(x+y)-xy(11)[/tex]

[tex]\implies 83=25(x+y)-11xy[/tex]

Multiply Equation 1 by 11:

[tex]\implies 275=121(x+y)-11xy[/tex]

Then subtract Equation 2 from this to eliminate 11xy and find the value of (x+y):

[tex]\implies 192=96(x+y)[/tex]

[tex]\implies (x+y)=2[/tex]

Multiply Equation 1 by 25:

[tex]\implies 625=275(x+y)-25xy[/tex]

Multiply Equation 2 by 11:

[tex]\implies 913=275(x+y)-121xy[/tex]

Subtract the 2nd from the 1st to eliminate 275(x+y) and find the value of xy:

[tex]\implies 288=-96xy[/tex]

[tex]\implies xy=-3[/tex]

Therefore, we now have:

Substitute these into the equation for ax⁵ + by⁵ and solve:

[tex]\implies ax^5+ by^5=(ax^4+by^4)(x+y)-xy(ax^3+by^3)[/tex]

[tex]\implies ax^5+ by^5=(83)(2)-(-3)(25)[/tex]

[tex]\implies ax^5+ by^5=166+75[/tex]

[tex]\implies ax^5+ by^5=241[/tex]

The test statistic is -0.46 and the p-value is gotten as 0.3340

The question is not complete because the data from the sample is 105, 107, 117, 106, 110.

Mean of sample is;

Mean = (105 + 107 + 117 + 106 + 110)/5

Mean; x' = 109

From online standard deviation calculator, the standard deviation of the sample is; S = 4.848

Now, the claim is that your team scored an average significantly less than 110 points. Thus, let's define the hypothesis as;

Null Hypothesis; μ = 110

Alternative Hypothesis; μ < 110

standard error of the mean; S_m = S/√n

S_m = 4.848/5

S_m = 2.17

Thus, the t-statistic is;

t = (x' - μ)/S_m

t = (109 - 110)/2.17

t = -0.46

Degree of freedom for this sample size is;

D.F = N - 1 =5 - 1 = 4

From online p-value from t-score calculator, we have;

p-value = 0.3340

The p-value is greater than the significance value and as such we fail to reject the null hypothesis and conclude that there is not enough evidence to support the claim that your team scored an average significantly less than 110 points.

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Green's theorem in vector calculus connects a line integral around a simple closed curve C to a double integral over the plane area D limited by C.

Green's theorem in vector calculus connects a line integral around a simple closed curve C to a double integral over the plane area D limited by C. It is a two-dimensional variant of Stokes' theorem.

The path of the particle is positively oriented so we can use Green's theorem to find the work done in moving particle along C.

The work done is given by [tex]\oint_{c}\vec F \cdot \vec{dr}[/tex]

Here, [tex]\vec F[/tex](x,y)=<x+y, y²-x>

Then [tex]\oint_{c}\vec F \cdot \vec{dr} = \oint_{c}(x+y)dx+(y^2-x)dy[/tex]

In comparison with [tex]\oint_{c}Pdx+Qdy[/tex] what we have,

P= x+y and Q=y²-x

Then [tex]\dfrac{\partial P}{\partial y} = 1[/tex] and [tex]\dfrac{\partial Q}{\partial x} =-1[/tex]

Using the Green's Theorem,

[tex]\oint_{c}Pdx+Qdy = \iint_{D} \left ( \dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right )dA[/tex]

Here D in polar Co-ordinates is given by,

D={(r,θ) : 0 ≤ r ≤ 7, 0 ≤ θ ≤ π}

Then

[tex]\oint_{c}(x+y)dx+(y^2-x)dy\\\\=-\iint_D 2dA\\\\=-\int_{\pi}^{0}\int_{0}^7 2r drd\theta\\\\=-49\pi[/tex]

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Answer:

g(x) = (x + 2)² + 1

Explanation:

Given parent function: f(x) = x²

If the function translates two units right

If the function translates one units up

Answer:

see explanation

Step-by-step explanation:

the set of x- values ( the input ) are called the domain

Answer:

domain

Step-by-step explanation:

as the y values is the range

Answer:

the remainder is 8

Step-by-step explanation:

2222 x 2223 x 22 x 24 = 2,608,059,168

2,608,059,168 ÷ 10 = 260,805,916.8

so the remainder is 8

Answer:

Radius = 20+5 = 25

We can find CE with the pythagorean theorem

20^2 + CE^2 = 25^2

CE = 15

The area of the shaded region = area of semicircle - area of triangle ACB = 11.32 cm².

Area of a triangle = 1/2(base)(height).

Area of a semi-circle = 1/2(πr²).

m∠ACB is a right triangle based on the central angle theorem. Therefore, ΔACB is a right triangle.

Radius of circle = OC = 4 cm

CB = 4 cm

Diameter AB = 2(4) = 8 cm

Using the Pythagorean theorem, find AC in ΔACB:

AC = √(AB² - CB²)

AC = √(8² - 4²)

AC ≈ 6.9 cm

Area of ΔACB = 1/2(CB)(AC)

Area of ΔACB = 1/2(4)(6.9)

Area of ΔACB ≈ 13.8 cm²

Area of semicircle = 1/2(πr²) = 1/2(π)(4²)

Area of semicircle ≈ 25.12 cm²

Area of the shaded region = area of semicircle - area of triangle ACB = 25.12 - 13.8

Area of the shaded region = 11.32 cm²

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[tex]x^4 +2x^3 -11x^2-12x+36\\ \\=x^4-2x^3+4x^3-8x^2-3x^2 +6x-18x+36\\\\=x^3(x-2) +4x^2(x-2) -3x(x-2) -18(x-2)\\\\=(x-2)(x^3 +4x^2 -3x -18)\\\\=(x-2)(x^3+3x^2+x^2+3x-6x-18)\\\\=(x-2)\textbf{[}x^2(x+3)+x(x+3)-6(x+3)\textbf{]}\\\\=(x-2)(x+3)(x^2 +x -6)\\\\=(x-2)(x+3)(x^2 +3x -2x-6)\\\\=(x-2)(x+3)\textbf{[}x(x+3) -2(x+3)\textbf{]}\\\\=(x-2)(x-2)(x+3)(x+3)\\\\=(x-2)^2(x+3)^2[/tex]

Answer:

27.5%

Step-by-step explanation:

Given:

Review:

Concepts:

Formula:

Now we must convert 275 milliliters to liters.

1. Divide the volume by 1,000.

275 ÷ 1000 = 0.275

2. Add the Unit Symbol for Liter.

0.275 ⇒ 0.275 L

Now we have to find how much of 1 liter- 0.275 liters as a percentage is.

To convert 0.275 to percent multiply 0.275 by 100. The result is 27.5 percent, or, using the percent sign, 27.5 %.

this is the answer hope it helps please if am wrong ask another person

thank you

Answer:

the answer is 14

Step-by-step explanation:

precalculus honors

There is sufficient evidence to conclude that, the percentage of the families who own a pet is different than 60%.

In null hypotheses, there is no relationship between the two phenomena under the assumption that it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.

It Is said That 60% of families own a pet.

Of a sample of 95 families, 70 owned pets.

Whether the percent of families who own pets is different than 60%.

Let P be a proportion of the family who owns pets

Then by the test, we have

H₀: P = 0.60

Hₐ: P ≠ 60

Then by the test statistic, we have

[tex]z_o = \dfrac{\bar{P} - P_o}{\sqrt{\dfrac{P_o(1 - P_o)}{n}}}[/tex]

Where

[tex]\bar P[/tex]  = sample proportion

P₀ = hypothesis proporion

n = sample size

Then we have

[tex]\bar P[/tex] = x/n

[tex]\bar P[/tex] = 70/95

[tex]\bar P[/tex] = 0.74

Then the test statistic will be

[tex]z_o = \dfrac{0.74-0.60}{\sqrt{\dfrac{0.60(1-0.60)}{95}}}[/tex]

z₀ = 2.785

Then the critical region will be

Critical value = ± [tex]z_{\alpha /2}[/tex]

α = 0.05

α/2 = 0.025

Then we have

z₀.₀₂₅ = 1.96

Then the critical value will be

Critical value = ± 1.96

We reject if [tex]|z_o| > |z_{\alpha /2}|[/tex]

We have

[tex]|z_o| > |z_{\alpha /2}|\\[/tex]

2.79 > 1.96

We reject the null hypothesis at a 5% significance level.

There is sufficient evidence to conclude that, the percentage of the families who own a pet is different than 60%.

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[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]

Let's evaluate ~

The given expression is :

[tex]\qquad \sf  \dashrightarrow \:4x + 8[/tex]

plug the value of x as 3

[tex]\qquad \sf  \dashrightarrow \:4(3) + 8[/tex]

[tex]\qquad \sf  \dashrightarrow \:12 + 8[/tex]

[tex]\qquad \sf  \dashrightarrow \:20[/tex]

Therefore, the required value is 20

Answer:

Yes, it is rational because it is a fraction. All rational numbers are either decimals or fractions. Since a fraction can also be represented as division, creating a decimal when divided, it counts as a rational number.

Answer:

[tex]x = -3~\text{and}~ x = -\dfrac 35[/tex]

Step-by-step explanation:

[tex]~~~~~~5x^2 +18x +9=0\\\\\implies 5x^2 +15x +3x +9 = 0\\\\\implies 5x(x+3) +3(x+3) = 0\\\\\implies (x+3)(5x+3) = 0\\\\\implies x = -3,~ x = -\dfrac 35[/tex]

Answer:

x = -3

OR

x = -(3/5)

Step by step explanation:

Given:

To Find:

Soln:

Use quadratic formulae:

[tex] \rm x=\cfrac{-b\pm\sqrt{b^2-4ac}}{2}[/tex]

According to the question,on the formula,

So substitute them on the formula:

THEN solve for x.

[tex] \rm \implies \: x = \cfrac{-1 8 \pm \sqrt{18 {}^{2} - 4(5 \times 9) }}{2 \times 5} [/tex]

[tex] \rm \implies \: x = \cfrac{-1 8 \pm \sqrt{324- 20 \times 9) }}{10} [/tex]

[tex]\rm \implies \: x = \cfrac{-1 8 \pm \sqrt{324- 180}}{10} [/tex]

[tex] \rm \implies \: x = \cfrac{ - 18 \pm \sqrt{144} }{10} [/tex]

[tex] \rm \implies \: x = \cfrac{ - 18 \pm12}{10}[/tex]

[tex] \rm \implies \: x = \cfrac{9 \pm 6}{5} [/tex]

Final solution will after adding first(9+6),then secondly subtracting both 9-6

[tex] \rm \implies \boxed{x = - \cfrac{3 }{5} }[/tex]

[tex] \implies \boxed{\rm x = - 3}[/tex]

So there'll be two possible answers.

Area of a Parallelogram = base x height

base = 3.3

height = 8.4

Area = 3.3 x 8.4 = 27.72

Answer: 27.72 cm^2

Hope this helps!

The order in which the two transformations in a composite transformation affect the final answer

To determine if order matters or not, I will use the following example and illustration.

Let the coordinate of point A be:

A = (2, 5)

Perform the following transformations:

First order: Translation before dilation

The translation by (x, y + 1) would give:

A' = (2, 5 + 1)

A' = (2,6)

The dilation by a scale factor of 2 would give

A" = 2 * (2,6)

A" = (4,12)

Second order: Dilation before translation

The dilation by a scale factor of 2 would give

A' = 2 * (2,5)

A' = (4,10)

The translation by (x, y + 1) would give:

A'' = (4, 10 + 1)

A'' = (4,11)

See that the final coordinates in both orders are not the same.

Hence, the order in which the two transformations in a composite transformation affect the final answer

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Answer:

hope this will help you

[tex]let \: the \: angle \: be \: \alpha \\ \cos( \alpha ) = \frac{b}{h} \\ \cos(60) = \frac{x}{9} \\ \frac{1}{2} = \frac{x}{9} \\ x = \frac{9}{2} \\ x = 4.5[/tex]

Answer:

$87.5

Step-by-step explanation:

Original price also means COST PRICE

C.P=$50

S.P=75% of $50-$50

To find the sale price it will be 75% of C.P -C.P

75% × $50 + $50= 0.75 × $50-$50

=$37.5-$50.

= $12.5

therefore the sale price is $12.5

THANK YOU

The price of the sweatshirt at the sale is $12.5 which is 75% off of the original price.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

Given, The original price of a sweatshirt is $50.

Also given that the sale price is 75% off the original price So, the cost of the sweatshirt is (100 - 75)% = 25% of the original price which is,

= (25/100)×50.

= $12.5.

So, The sale price of the sweatshirt is $12.5.

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Answer:

a. 10.

Step-by-step explanation:

27 / 3 = 9

90/10 = 9

So it's 10.

Answer:

15

Step-by-step explanation:

First lets simplify the left side :

5/12 + 5/12 + 5/12 = 15/12

15/12

To make the right side into 15/12 we must multiply it by 15:

15 × 1/12 = 15/12

So the blank is 15

Hope this helped and brainliest please.

Answer:

15/1

Step-by-step explanation:

you simply add the 5's only not the 12. when u multiply 1 and 12 u get 12 and when you multiply 15 and 1 you get 15

Answer:

y = -(2x+16) / 5

Step-by-step explanation:

2x + 5y = -16

5y = -2x - 16

y = (-2x - 16) / 5

y = -(2x+16) / 5

The value of a car after 10 years be $20898.27 if the car rental company assumes each car in their fleet depreciates by 6% per year option (a) $20898.27 is correct.

It is defined as the function that rapidly increases and the value of the exponential function is always a positive. It denotes with exponent [tex]\rm y = a^x[/tex]

where a is a constant and a>1

We can solve this problem by exponential function:

The word  depreciate means the price is decreasing.

We can find the value be when the car is 10 years old:

p = 38800(1 - 0.06)¹⁰

p = 38800(0.94)¹⁰

p = 20898.266 ≈ $20898.27

Thus, the value of a car after 10 years be $20898.27 if the car rental company assumes each car in their fleet depreciates by 6% per year option (a) $20898.27 is correct.

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The probability that the next customer will pay with a credit card is 0.61.

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it.

Given that Cristian's customers pay using, cash, debit cards and credit cards,

He knows that in the last week 10 customers paid cash, 14 customers used a debit card, and 38 customers used a credit card.

We need to find the probability that the next customer will pay with a credit card,

Therefore,

The favorable outcomes = 38

Total outcomes = 38+10+14 = 62

Therefore,

P(credit cards) = 38/62 = 0.61

Hence the probability that the next customer will pay with a credit card is 0.61.

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The given sum for n = 1 to n = 15 is equal to 255, so the correct option is B.

We want to find the sum of the series whose elements are of the form:

(2n + 1)

From n = 1 to n = 15.

Then our sum will be:

(2*1 + 1) + (2*2 + 1) + ... + (2*15 + 1).

3 + 5 + ... + 31

This is the sum of all odd numbers in the interval [3, 31]

Which gives:

[tex]S = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 = 255[/tex]

So the correct option is B.

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