find the value of the trigonometric ratio. make sure to simplify the fraction if needed

The hypotenuse will always be the longest side of the triangle. Option C is correct: AB > DC.

AB is the hypotenuse of triangle ABC. Therefore, it is greater than leg AC. AC is the hypotenuse of triangle ACD. If AC is less than AB, then DC must also be less than AB because DC is less than AC.

Hope this helps!

Answer:

x = 1/5

Step-by-step explanation:

5(3x + 4) = 23

Distribute

15x+20 = 23

Subtract 20 from each side

15x+20-20 = 23-20

15x = 3

Divide by 15

15x/15 = 3/15

x = 1/5

Answer:

[tex]x = 0.2[/tex]

Step-by-step explanation:

Step 1:  Distribute

[tex]5(3x + 4) = 23[/tex]

[tex](5 * 3x) + (5 * 4) = 23[/tex]

[tex](15x) + 20 = 23[/tex]

Step 2:  Subtract 20 from both sides

[tex](15x) + 20 - 20 = 23 - 20[/tex]

[tex]15x = 3[/tex]

Step 3:  Divide both sides by 15

[tex]\frac{15x}{15} = \frac{3}{15}[/tex]

[tex]x = \frac{1}{5}[/tex]

[tex]x = 0.2[/tex]

Answer:  [tex]x = 0.2[/tex]

Answer:

The rate at which the distance between the two cars is increasing is 30 mi/h

Step-by-step explanation:

Given;

speed of the first car, v₁ = 24 mi/h

speed of the second car, v₂ = 18 mi/h

Two hours later, the position of the cars is calculated as;

position of the first car, d₁ = 24 mi/h  x 2 h  = 48 mi

position of the second car, d₂ = 18 mi/h  x 2 h = 36 mi

The displacement of the two car is calculated as;

displacement, d² = 48² + 36²

                        d² = 3600

                        d = √3600

                        d = 60 mi

The rate at which this displacement is changing = (60 mi) / (2h)

                                                                                 = 30 mi/h

Answer:

the probability that a truck drives less than 159 miles in a day = 0.9374

Step-by-step explanation:

Given;

mean of the truck's speed, (m) = 120 miles per day

standard deviation, d = 23 miles per day

If the mileage per day is normally distributed, we use the following conceptual method to determine the probability of less than 159 miles per day;

1 standard deviation above the mean = m + d, = 120 + 23 = 143

2 standard deviation above the mean = m + 2d, = 120 + 46 = 166

159 is below 2 standard deviation above the mean but greater than 1 standard deviation above the mean.

For normal districution, 1 standard deviation above the mean = 84 percentile

Also, 2 standard deviation above the mean = 98 percentile

143 --------> 84%

159 ---------> x

166 --------- 98%

[tex]\frac{159-143}{166-143} = \frac{x-84}{98-84} \\\\\frac{16}{23} = \frac{x-84}{14} \\\\23(x-84) = 224\\\\x-84 = 9.7391\\\\x = 93.7391\ \%[/tex]

Therefore, the probability that a truck drives less than 159 miles in a day = 0.9374

I think the given series is

[tex]\displaystyle\sum_{n=2}^\infty \frac{n^2}{n^3+1}[/tex]

You can use the integral test because the summand is clearly positive and decreasing. Then

[tex]\displaystyle\sum_{n=2}^\infty\frac{n^2}{n^3+1} > \int_2^\infty\frac{x^2}{x^3+1}\,\mathrm dx[/tex]

Substitute u = x ³ + 1 and du = 3x ² dx, so the integral becomes

[tex]\displaystyle \int_2^\infty\frac{x^2}{x^3+1}\,\mathrm dx = \frac13\int_9^\infty\frac{\mathrm du}u = \frac13\ln(u)\bigg|_{u=9}^{u\to\infty}[/tex]

As u approaches infinity, we have ln(u) also approaching infinity (whereas 1/3 ln(9) is finite), so the integral and hence the sum diverges.

Answer:

The price that maximizes the revenue is $165

Step-by-step explanation:

We can model the price as a function of sold units as a linear relationship.

Remember that a linear relationship is something like:

y = a*x + b

where a is the slope and b is the y-intercept.

We know that if the line passes through the points (x₁, y₁) and (x₂, y₂), then the slope can be written as:

a = (y₂ - y₁)/(x₂ - x₁)

For this line, we have the point (40, $110)

which means that to sell 40 units, the price must be $110

And we know that if the price increases by $11, then he will sell 2 units less.

Then we also have the point (38, $121)

So we know that our line passes through the points (40, $110) and  (38, $121)

Then the slope of the line is:

a = ($121 - $110)/(38 - 40) = $11/-2 = -$5.5

Then the equation of the line is:

p(x) = -$5.5*x + b

to find the value of b, we can use the point (40, $110)

This means that when x = 40, the price is $110

then:

p(40) = $110 = -$5.5*40 + b

            $110 = -$220 + b

       $110 + $220 = b

        $330 = b

Then the price equation is:

p(x) = -$5.5*x + $330

Now we want to find the maximum revenue.

The revenue for selling x items, each at the price p(x), is:

revenue = x*p(x)

replacing the p(x) by the equation we get:

revenue = x*(-$5.5*x + $330)

revenue = -$5.5*x^2 + $330*x

Now we want to find the x-value for the maximum revenue.

You can see that the revenue equation is a quadratic equation with a negative leading coefficient. This means that the maximum is at the vertex.

And remember that for a quadratic equation like:

y = a*x^2 + b*x + c

the x-value of the vertex is:

x = -b/2a

Then for our equation:

revenue = -$5.5*x^2 + $330*x

the x-vale of the vertex will be:

x = -$330/(2*-$5.5) = 30

x = 30

This means that the revenue is maximized when we sell 30 units.

And the price is p(x) evaluated in x = 30

p(30) = -$5.5*30 + $330 = $165

The price that maximizes the revenue is $165

Answer:

there are infinite solutions

Step-by-step explanation:

if you add y-3 to both sides of the first equation, you will see that it is equal to the second equation, so they are the same line. Therefore, there are infinite solutions to this system

Answer:

Answer : 1/8.

Step-by-step explanation:

Hey there!

Please see the attached picture for your answer.

Hope it helps!

Answer:

[tex]x^\frac{3}{5} = (x^3 )^\frac{1}{5}[/tex]

[tex]x^\frac{3}{5} = \sqrt[5]{x^3}[/tex]

[tex]x^\frac{3}{5} = (\sqrt[5]{x})^3[/tex]

Step-by-step explanation:

Given

[tex]x^\frac{3}{5}[/tex]

Required

The equivalent expressions

We have:

[tex]x^\frac{3}{5}[/tex]

Expand the exponent

[tex]x^\frac{3}{5} = x^{ 3 * \frac{1}{5}}[/tex]

So, we have:

[tex]x^\frac{3}{5} = (x^3 )^\frac{1}{5}[/tex] ----- this is equivalent

Express 1/5 as roots (law of indices)

[tex]x^\frac{3}{5} = \sqrt[5]{x^3}[/tex] ------ this is equivalent

The above can be rewritten as:

[tex]x^\frac{3}{5} = (\sqrt[5]{x})^3[/tex] ------ this is equivalent

9514 1404 393

Answer:

  x = 14 cm

Step-by-step explanation:

We can only solve for x if the triangles are similar. The arrows on the left and right legs say those are parallel. Since alternate interior angles at each of the transversals are congruent, the triangles are AA similar.

ΔABC ~ ΔDEC, so we have ...

  EC/ED = BC/BA

  x/(18 cm) = (35 cm)/(45 cm)

  x = (18 cm)(7/9) = 14 cm

Answer:

The zeroes of the function are: 3 and 4

Step-by-step explanation:

Step-by-step explanation:

Area ABC=1/2ab×sin C

=1/2 ×20×10 ×Sin C

Sin C =100

Answer:

how many times should u use the metal rod ??

Answer:

Break even sales = $17,500 (Approx.)

Step-by-step explanation:

Given:

Fixed expenses = $7,000

Contribution ratio = 40%

Find:

Break even sales dollars

Computation:

Break even sales = Fixed expenses / Contribution ratio

Break even sales = 7,000 / 40%

Break even sales = 7,000 / 0.40

Break even sales = 17,500

Break even sales = $17,500 (Approx.)

Answer:

obtuse                    

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Answer:

x = 5 and -6

Step-by-step explanation:

Using the intersecting chord theorem which states that the products of the lengths of the line segments on each chord are equal.

Hence:

let

a = 6, b = 5, c = x+1 and d = x

Therefore, ab = cd

6*5 = x(x+1)

30 = x²+x

x²+x - 30 = 0

x²+ 6x - 5x - 30 = 0

x(x+6) - 5(x+6) =0

(x-5)(x+6) = 0

x-5 =0 and x+6 = 0

x = 5 and -6

Answer:

17 units

Step-by-step explanation:

Use the distance formula, [tex]d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]

Plug in the points and simplify:

[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}[/tex]

[tex]d = \sqrt{(-8 - 7)^2 + (0-8)^2}[/tex]

[tex]d = \sqrt{(-15)^2 + (-8)^2}[/tex]

[tex]d = \sqrt{225 + 64[/tex]

[tex]d = \sqrt{289[/tex]

[tex]d = 17[/tex]

So, the distance between the points is 17 units

Step-by-step explanation:

1. In an arithmetic sequence, the general term can be written as

xₙ = y + d(a-1), where xₐ represents the ath term, y is the first value, and d is the common difference.

Given the third term and the fifth term, and knowing that the difference between each term is d, we can say that the 4th term is x₃+d and the fifth term is the fourth term plus d, or (x₃+d)+d =

x₃+2d. =x₅ Given x₃ and x₅, we can subtract x₃ from both sides to get

x₅-x₃ = 2d

divide by 2 to isolate d

(x₅-x₃)/2 = d

This lets us solve for d. Given d, we can say that

x₃ = y+d(2)

subtract 2*d from both sides to isolate the y

x₃ -2*d = y

Therefore, because we know x₃ and d at this point, we can solve for y, letting us plug y and d into our original equation of

xₙ = y + d(a-1)

2.

Given the third and fifth term, with a common ratio of r, we can say that the fourth term is x₃ * r. Then, the fifth term is

x₃* r * r

= x₃*r² = x₅

divide both sides by x₃ to isolate the r²

x₅/x₃ = r²

square root both sides

√(x₅/x₃) = ±r

One thing that is important to note is that we don't know whether r is positive or negative. For example, if x₃ = 4 and x₅ = 16, regardless of whether r is equal to 2 or -2, 4*r² = 16. I will be assuming that r is positive for this question.

Given the common ratio, we can find x₆ as x₅ * r, x₇ as x₅*r², and all the way up to x₁₀ = x₅*r⁵. We don't know the general term, but can still find the tenth term of the sequence

= 2025

When you are told to find the smallest length possible, you perform L.C.M(Least common multiples)

For this, you divide the given lengths using the numbers that divides all through.

I have added an image to this answer. Go through it for more explanation

Answer:

Our two intersection points are:

[tex]\displaystyle (3, -2) \text{ and } \left(-\frac{53}{25}, \frac{46}{25}\right)[/tex]

Step-by-step explanation:

We want to find where the two graphs given by the equations:

[tex]\displaystyle (x+1)^2+(y+2)^2 = 16\text{ and } 3x+4y=1[/tex]

Intersect.

When they intersect, their x- and y-values are equivalent. So, we can solve one equation for y and substitute it into the other and solve for x.

Since the linear equation is easier to solve, solve it for y:

[tex]\displaystyle y = -\frac{3}{4} x + \frac{1}{4}[/tex]

Substitute this into the first equation:

[tex]\displaystyle (x+1)^2 + \left(\left(-\frac{3}{4}x + \frac{1}{4}\right) +2\right)^2 = 16[/tex]

Simplify:

[tex]\displaystyle (x+1)^2 + \left(-\frac{3}{4} x + \frac{9}{4}\right)^2 = 16[/tex]

Square. We can use the perfect square trinomial pattern:

[tex]\displaystyle \underbrace{(x^2 + 2x+1)}_{(a+b)^2=a^2+2ab+b^2} + \underbrace{\left(\frac{9}{16}x^2-\frac{27}{8}x+\frac{81}{16}\right)}_{(a+b)^2=a^2+2ab+b^2} = 16[/tex]

Multiply both sides by 16:

[tex](16x^2+32x+16)+(9x^2-54x+81) = 256[/tex]

Combine like terms:

[tex]25x^2+-22x+97=256[/tex]

Isolate the equation:

[tex]\displaystyle 25x^2 - 22x -159=0[/tex]

We can use the quadratic formula:

[tex]\displaystyle x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

In this case, a = 25, b = -22, and c = -159. Substitute:

[tex]\displaystyle x = \frac{-(-22)\pm\sqrt{(-22)^2-4(25)(-159)}}{2(25)}[/tex]

Evaluate:

[tex]\displaystyle \begin{aligned} x &= \frac{22\pm\sqrt{16384}}{50} \\ \\ &= \frac{22\pm 128}{50}\\ \\ &=\frac{11\pm 64}{25}\end{aligned}[/tex]

Hence, our two solutions are:

[tex]\displaystyle x_1 = \frac{11+64}{25} = 3\text{ and } x_2 = \frac{11-64}{25} =-\frac{53}{25}[/tex]

We have our two x-coordinates.

To find the y-coordinates, we can simply substitute it into the linear equation and evaluate. Thus:

[tex]\displaystyle y_1 = -\frac{3}{4}(3)+\frac{1}{4} = -2[/tex]

And:

[tex]\displaystyle y _2 = -\frac{3}{4}\left(-\frac{53}{25}\right) +\frac{1}{4} = \frac{46}{25}[/tex]

Thus, our two intersection points are:

[tex]\displaystyle (3, -2) \text{ and } \left(-\frac{53}{25}, \frac{46}{25}\right)[/tex]

Answer:

c. The researchers are 95% confident that the true mean difference in ALT values between the population of drinkers and population of non-drinkers is between 24.32 and 39.68.

Step-by-step explanation:

Given :

Groups:

x1 = 69 ; s1 = 19 ; n1 = 38

x2 = 32 ; s2 = 14 ; n2 = 37

1 - α = 1 - 0.95 = 0.05

Using a confidence interval calculator to save computation time, kindly plug the values into the calculator :

The confidence interval obtained is :

(24.32 ; 39.68) ; This means that we are 95% confident that the true mean difference in ALT values between the two population lies between

(24.32 ; 39.68) .

Answer:

D) 0.89

Step-by-step explanation:

round 0.885 to 0.89

Let it be x

[tex]\\ \sf\longmapsto \dfrac{3}{5}=\dfrac{x}{x+4}[/tex]

[tex]\\ \sf\longmapsto 3(x+4)=5x[/tex]

[tex]\\ \sf\longmapsto 3x+12=5x[/tex]

[tex]\\ \sf\longmapsto 5x-3x=12[/tex]

[tex]\\ \sf\longmapsto 2x=12[/tex]

[tex]\\ \sf\longmapsto x=\dfrac{12}{2}[/tex]

[tex]\\ \sf\longmapsto x=6[/tex]

Answer:

Step-by-step explanation:

Answer:

The original height of the plant was 2 mm

Step-by-step explanation:

Given

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

Required

Interpret the y-intercept

The y-intercept is when [tex]x = 0[/tex]

So, we have:

[tex]y = 2 + 4 *0[/tex]

[tex]y = 2 + 0[/tex]

[tex]y = 2[/tex]

This implies that the original or initial height was 2 mm

Answer:

isosceles acute

Step-by-step explanation:

We have two sides that are equal so the triangle is isosceles

None of the angles are larger than 90 degrees so non of the angles are obtuse,  and none of the angles are 90 degrees, so the triangle is acute ( which means the angles are all less than 90 degrees)

Answer:

A

Step-by-step explanation:

The graph is a square root function