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2 years ago

What are the values of a 1 and r of the geometric series? 1 + 3 + 9 + 27 + 81

Mathematics

2 answers:

hammer [34]2 years ago

4 0

Answer:

a₁ = 1; r = 3

Step-by-step explanation:

a₁ represents the first term. It is 1 in this case.

r represents the common ratio between terms. In this case it's 3 / 1 = 3.

zysi [14]2 years ago

3 0

Answer:

a1 = 1

r = 3

Step-by-step explanation:

a1 = first term = 1

r = common ratio = 2nd term/1st term

= 3/1

r = 3

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4. Nine fruit juice bottles contain 4 1/2 litres of fruit juice between them.

brilliants [131]

Answer:

2 $\frac{1}{2}$ litres or 2.5 litres

Step-by-step explanation:

From the question, 9 fruit juice bottles contain 4 $\frac{1}{2}$ litres of fruit juice. Therefore, to find how many litres of fruit juice would be contained in 5 fruit juice bottles, you multiply 5 by 4 $\frac{1}{2}$ litres and you divide the product by 9.

i.e. Let the quantity of the fruit juice for 5 bottles be x.

9 juice bottles ===> 4 $\frac{1}{2}$ litres

5 juice bottles ===> x

∴x = $\frac{5 * 4\frac{1}{2} }{9} litres$

x = $\frac{5 * 4.5}{9} litres$

x = $\frac{22.5}{9} litres$

x = 2.5 litres or 2 $\frac{1}{2}$ litres

Hope this helps!!!

3 0

2 years ago

(5x+2) and {3(x+14)} what is

Phoenix [80]

Answer:

Step-by-step explanation:

x=20

Reorder the terms:

(2 + 5x) = 3(x + 14)

Remove parenthesis around (2 + 5x)

2 + 5x = 3(x + 14)

Reorder the terms:

2 + 5x = 3(14 + x)

2 + 5x = (14 * 3 + x * 3)

2 + 5x = (42 + 3x)

Solving

2 + 5x = 42 + 3x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3x' to each side of the equation.

2 + 5x + -3x = 42 + 3x + -3x

Combine like terms: 5x + -3x = 2x

2 + 2x = 42 + 3x + -3x

Combine like terms: 3x + -3x = 0

2 + 2x = 42 + 0

2 + 2x = 42

Add '-2' to each side of the equation.

2 + -2 + 2x = 42 + -2

Combine like terms: 2 + -2 = 0

0 + 2x = 42 + -2

2x = 42 + -2

Combine like terms: 42 + -2 = 40

2x = 40

Divide each side by '2'.

x = 20

Simplifying

x = 20

7 0

2 years ago

Can someone please help!!

leva [86]

Answer:

Step-by-step explanation:

the last one

8 0

2 years ago

A metal cylinder can with an open top and closed bottom is to have volume 4 cubic feet. Approximate the dimensions that require

Aleksandr-060686 [28]

Answer:

$r\approx 1.084\ feet$

$h\approx 1.084\ feet$

$\displaystyle A=11.07\ ft^2$

Step-by-step explanation:

<u>Optimizing With Derivatives </u>

The procedure to optimize a function (find its maximum or minimum) consists in :

Produce a function which depends on only one variable
Compute the first derivative and set it equal to 0
Find the values for the variable, called critical points
Compute the second derivative
Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum

We know a cylinder has a volume of 4 $ft^3$ . The volume of a cylinder is given by

$\displaystyle V=\pi r^2h$

Equating it to 4

$\displaystyle \pi r^2h=4$

Let's solve for h

$\displaystyle h=\frac{4}{\pi r^2}$

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

$\displaystyle A=\pi r^2+2\pi rh$

Replacing the formula of h

$\displaystyle A=\pi r^2+2\pi r \left (\frac{4}{\pi r^2}\right )$

Simplifying

$\displaystyle A=\pi r^2+\frac{8}{r}$

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

$\displaystyle A'=2\pi r-\frac{8}{r^2}=0$

Rearranging

$\displaystyle 2\pi r=\frac{8}{r^2}$

Solving for r

$\displaystyle r^3=\frac{4}{\pi }$

$\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet$

Computing h

$\displaystyle h=\frac{4}{\pi \ r^2}\approx 1.084\ feet$

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

$\displaystyle A''=2\pi+\frac{16}{r^3}$