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3 years ago

Please simplify: −23d+81≤−98d+1

1 answer:

5 0

$- 23 d + 81 \leq -98 d + 1$

Subtract 81 from both sides :

$-23 d + 81-81 \leq -98d+1 -81$

$-23d \leq -98d-80$

Add 98d to both sides:

$-23d +98 d \leq -98d-80+98d$

$75d \leq -80$

Divide both sides by 75 :

$\frac{75d}{75} \leq \frac{-80}{75}$

$d \leq =- \frac{16}{15}$

hope this helps!.

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Equilateral triangle ABC has an area of \sqrt{3}√ 3 . If the shaded region has an area of \piπK − \sqrt{3}√ 3 , what

Liono4ka [1.6K]

Answer:

The value of k = 4/3

Step-by-step explanation:

* Lets explain how to solve the problem

- An equilateral triangle ABC is inscribed in a circle N

- The area of the triangle is √3

- The shaded area is the difference between the area of the circle

and the area of the equilateral triangle ABC

- The shaded are = k π - √3

- We need to find the value of k

* At first lets find the length of the side of the Δ ABC

∵ Δ ABC is an equilateral triangle

∴ Its area = √3/4 s² , where s is the length of its sides

∵ The area of the triangle = √3

∴ √3/4 s² = √3

- divide both sides by √3

∴ 1/4 s² = 1

- Multiply both sides by 4

∴ s² = 4 ⇒ take √ for both sides

∴ s = 2

∴ The length of the side of the equilateral triangle is 2

* Now lets find the radius of the circle

- In the triangle whose vertices are A , B and N the center of the circle

∵ AN and BN are radii

∴ AN = BN = r , where r is the radius of the circle

∵ The sides of the equilateral angles divides the circle into 3 equal

arcs in measure where each arc has measure 360°/3 = 120°

∵ The measure of the central angle in a circle equal the measure

of the its subtended arc arc

∵ ∠ANB is an central angle subtended by arc AB

∵ The measure of arc AB is 120°

∴ m∠ANB = 120°

- By using the cosine rule in Δ ANB

∵ AB = 2 , AN = BN = r , m∠ANB = 120°

∴ $(2)^{2}=r^{2}+r^{2}-2(r)(r)cos(120)$

∴ $4=r^{2}+r^{2}-2(r)(r)(-0.5)$

∴ $4=r^{2}+r^{2}-(-r^{2})$

∴ $4=r^{2}+r^{2}+r^{2}$

∴ $4=3r^{2}$

- Divide both sides by 3

∴ $r^{2}=\frac{4}{3}$

- Take square root for both sides

∴ r = 2/√3

* Lets find the value of k

∵ Area circle = πr²

∵ r = 2/√3

∴ Area circle = π(2/√3)² = (4/3)π

∵ Area shaded = area circle - area triangle

∵ Area triangle = √3

∴ Area shaded = (4/3) π - √3

∵ Area of the shaded part is π k - √3

- Equate the two expressions

∴ π k - √3 = (4/3) π - √3

∴ k = 4/3

* The value of k = 4/3

7 0

3 years ago

Ethan has 3 pounds of mixed nuts. How many ounces of mixed nuts does Ethan have?

almond37 [142]

There are 16 ounces in a pound so all you have to do is multiply 16 and 3.
16*3=48
ANSWER= 48 ounces

5 0

3 years ago

“encontrar la integral indefinida y verificar el resultado mediante derivación”

Oliga [24]

$I=\displaystyle\int\frac x{(1-x^2)^3}\,\mathrm dx$

Haz la sustitución:

$y=1-x^2\implies\mathrm dy=-2x\,\mathrm dx$

$\implies I=\displaystyle-\frac12\int\frac{\mathrm dy}{y^3}=\frac1{4y^2}+C=\frac1{4(1-x^2)^2}+C$

Para confirmar el resultado:

$\dfrac{\mathrm dI}{\mathrm dx}=\dfrac14\left(-\dfrac{2(-2x)}{(1-x^2)^3}\right)=\dfrac x{(1-x^2)^3}$

$I=\displaystyle\int\frac{x^2}{(1+x^3)^2}\,\mathrm dx$

Sustituye:

$y=1+x^3\implies\mathrm dy=3x^2\,\mathrm dx$

$\implies I=\displaystyle\frac13\int\frac{\mathrm dy}{y^2}=-\frac1{3y}+C=-\frac1{3(1+x^3)}+C$

(Te dejaré confirmar por ti mismo.)

$I=\displaystyle\int\frac x{\sqrt{1-x^2}}\,\mathrm dx$

Sustituye:

$y=1-x^2\implies\mathrm dy=-2x\,\mathrm dx$

$\implies I=\displaystyle-\frac12\int\frac{\mathrm dy}{\sqrt y}=-\frac12(2\sqrt y)+C=-\sqrt{1-x^2}+C$

$I=\displaystyle\int\left(1+\frac1t\right)^3\frac{\mathrm dt}{t^2}$

Sustituye:

$u=1+\dfrac1t\implies\mathrm du=-\dfrac{\mathrm dt}{t^2}$

$\implies I=-\displaystyle\int u^3\,\mathrm du=-\frac{u^4}4+C=-\frac{\left(1+\frac1t\right)^4}4+C$

Podemos hacer que esto se vea un poco mejor:

$\left(1+\dfrac1t\right)^4=\left(\dfrac{t+1}t\right)^4=\dfrac{(t+1)^4}{t^4}$

$\implies I=-\dfrac{(t+1)^4}{4t^4}+C$

4 0

3 years ago

Elisa is making candles she knows each candle needs a wick that is 4 inches long.she wants to make 7 candles how many inches of

dimulka [17.4K]

She will need 28 inches of wick

4 0

3 years ago

Read 2 more answers

Factor the trinomial step by step & explain 2x^2+ 5x + 3

Colt1911 [192]

Answer:

One way to factor equation is to find the zeros. Its obvious that x=-1 is solution for this. So one factor is (x+1)

the next factor should include 2x at first (because we have 2x^2 in the equation which can not be made any other way)

Let's suppose the factor is (x+1)(2x+b)

Since we do not know what is b open the brackets 2x^2+bx+2x+b. If this is equal to 2x^2+5x+3 then b=3. We are left with (x+1)(2x+3)=2(x+1)(x+3/2)

Would go for other shorter solutions, but they require some deeper understanding like Horner's scheme,Fermat's Theorem or even deeper which I assume You will not understand)

8 0

3 years ago