What is the sum of the geometric series in which a1 = 4, r = 3, and an = 324?
Hint: cap s sub n equals start fraction a sub one left parenthesis one minus r to the power of n end power right parenthesis over one minus r end fraction comma r ≠ 1, where a1 is the first term and r is the common ratio.

The interpretation of v(4) = 64 for a cube's volume means a cube with 4-unit sides has a 64 cubic unit volume. Doubling the sides to 8 units leads to a volume of 512 cubic units, resulting in a volume ratio of 8:1 because volume scales with the cube of the side length increase.

The statement v(4) = 64 indicates that a cube with side length 4 units has a volume of 64 cubic units. When each side of the cube is doubled, the new side length becomes 8 units. The volume of a cube is calculated as the cube of its side length (s³). So, the new volume is 8³ or 512 cubic units.

The ratio of the new volume to the old volume is 512:64, which simplifies to 8:1. This ratio is not simply 2 because the volume of a cube increases by the cube of the factor by which the side is increased. Since the sides are doubled (a factor of 2), the volume increases by 2³ or 8 times, not 2 times.

This reasoning outlines why the ratio of the volumes of two cubes, when one cube has sides twice as long as the other, is not 2 but 8:1. The surface area to volume ratio is also affected by changes in the side length, which is significant not only in geometry but also in disciplines such as biology.

Answer:

96

Step-by-step explanation:

All you gotta do is multiply 7 by a high number like say 80 you wont get 672 just keep on multiplying it by higher numbers till you get 672 i hope i helped

Answer with explanation:

The function for which we have to find , the intervals on which the polynomial is entirely negative and those on which it is entirely positive.

 f(x)= -x²+6 x -10

 If you will find the root of the function, there is no real root.

To find the root we will use Discriminant formula

For a Quadratic function, ax²+b x+c=0,

 [tex]x=\frac{-b+\sqrt{D}}{2a}\\\\D=b^2-4ac\\\\x=\frac{-6\pm \sqrt{6^2-4 *(-1)*(-10)}}{2*(-1)}\\\\x=\frac{-6\pm\sqrt{-4}}{-2}[/tex]

→So,there is no interval in which  the polynomial is entirely negative and those on which it is entirely positive, which can be represented in interval notation using Ф.

   [tex]f(x)= -(x^2-6x+10)\\\\= - [(x-3)^2-9+10]\\\\=-(x-3)^2-1[/tex]

For, any value of x,the value of f(x) will be always Negative.

To find the vertex, put ,x-3=0

 x=3

And, by putting , x= 3 ,in the equation we get

y = -1

So,Vertex = (3, -1)

⇒If you will try to find the intervals in which the function is increasing , means the curve is moving up is from (-∞, 3) and the intervals in which the function is decreasing , means the curve is moving downward is from , (3, ∞).

Answer:  The required solution for b is [tex]b=\dfrac{2c+4}{3}.[/tex]

Step-by-step explanation:  We are given to solve the following equation for b :

[tex]\dfrac{3b-4}{2}=c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

To solve the given equation for b, we must take b on one side of the equation and all other terms on the other side.

From equation (i), we get

[tex]\dfrac{3b-4}{2}=c\\\\\Rightarrow 3b-4=2c\\\\\Rightarrow 3b=2c+4\\\\\Rightarrow b=\dfrac{2c+4}{3}.[/tex]

Thus, the required solution for b is [tex]b=\dfrac{2c+4}{3}.[/tex]

48:3= 16  36:2=18  96:6= 16

To find the product of the expression (a-3/7) divided by (3-a/21), multiply (a-3/7) by the reciprocal of (3-a/21), yielding the simplified expression ((21a - 9)/ (21a - 7)) valid for all values of 'a' except when 'a' equals 8/3.

The student is asking for the product of the expression (a-3/7) divided by (3-a/21). To simplify this expression, we need to apply the division of fractions rule, which states that to divide by a fraction, you multiply by its reciprocal. The reciprocal of (3-a/21) is (21/(3a-1)). So our new expression is (a-3/7) × (21/(3a-1)).

We can multiply the numerators and denominators separately: (a-3/7) × 21 = (a× 21 - 3× 3) and 7 × (3a-1) = 21a-7. Simplified, we get ((21a - 9)/ (21a - 7)) as the product.

However, we should also check if the expression is defined for all values of 'a' as there may be values for which the denominator becomes zero. Specifically, we need 'a' to not be such that (3a-1) equals 7, which simplifies to a = 8/3. So for all values of 'a' except when 'a' equals 8/3, the product of the given expression is ((21a - 9)/ (21a - 7)).

Final Answer:

La cantidad de agua que se debe agregar para que haya 20 gramos de cal por litro es de aproximadamente 0.625 litros. Esto se calcula ajustando la proporción actual de 250 gramos de cal en 8 litros de agua. La nueva proporción será de 20 gramos de cal por litro.

Step-by-step explanation:

Para resolver este problema, podemos usar una regla de tres simple. La relación actual es de 250 gramos de cal por 8 litros de agua. Queremos saber cuántos gramos de cal habría en 1 litro de agua.

Primero, calculemos cuántos gramos de cal hay por litro de agua en la situación actual:

[tex]\( \frac{250 \, \text{gramos de cal}}{8 \, \text{litros de agua}} \)[/tex]

Ahora, queremos encontrar cuántos gramos de cal habría en 1 litro de agua, así que multiplicamos la cantidad actual por 1/8:

[tex]\( \frac{250 \, \text{gramos de cal}}{8 \, \text{litros de agua}} \times \frac{1 \, \text{litro de agua}}{8} \)[/tex]

Esto nos dará la cantidad de gramos de cal por litro de agua en la nueva situación. Ahora, queremos que esta cantidad sea de 20 gramos:

[tex]\( \frac{250 \, \text{gramos de cal}}{8 \, \text{litros de agua}} \times \frac{1 \, \text{litro de agua}}{8} = x \, \text{gramos de cal por litro} \)[/tex]

Queremos que ( x ) sea igual a 20 gramos:

[tex]\( x = 20 \, \text{gramos de cal por litro} \)[/tex]

Entonces, puedes resolver esta ecuación para encontrar cuántos litros de agua se necesitan:

[tex]\( \frac{250 \, \text{gramos de cal}}{8 \, \text{litros de agua}} \times \frac{1 \, \text{litro de agua}}{8} = 20 \, \text{gramos de cal por litro} \)[/tex]

Resolviendo esta ecuación, obtendrás la cantidad de litros de agua necesarios para lograr que haya 20 gramos de cal por litro.

Answer: [tex]p(x)= 13x-20[/tex]

Step-by-step explanation:

Given: The revenue for mowing x lawns is [tex]r(x) = 18x[/tex] .

Becca's cost for gas and the mower rental is c(x) = 5x + 20

The profit from mowing x lawns is given by :-

[tex]p(x)=(r-c)(x)[/tex]

The above  function can be written as

[tex]p(x)=(r-c)(x)=r(x)-c(x)\\\\\Rightarrow p(x)= 18x-(5x+20)\\\\\Rightarrow p(x)= 18x-5x-20\\\\\Rightarrow p(x)= 13x-20[/tex]

Answer:

Average height increase for each additional centimeter of armspan is 0.93cm

What is height increase ?

Change in height from the last one is the increase of height .

Given , height y = 6.4 + 0.93x

            For x=1

              y1 = 6.4+ 0.93

            For x=2

              y2 = 6.4 + 0.93*2

            For x=3

              y3 = 6.4 + 0.93*3

   Here height increase

              y2-y1 = 0.93

              y3-y2 = 0.93

Hence,  average height increase for each additional centimeter of armspan is 0.93cm.

To learn more about finding average , click :

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Answer:

213 euros per month

Step-by-step explanation:

1dollar = 0.7306euros

thus $3,500 dollars can be found by a rule of three

dollar             euro

1             ⇒  0.7306

3500     ⇒        x

we find x as follows:

[tex]x=\frac{0.7306*3500}{1} = 2,557.1[/tex] euros

She will pay 2,557.1  euros per year.

Dividing between the 12 months of the year:

[tex]\frac{2,557.1}{12}=213.1[/tex]

Rounding to the nearest whole euro she will pay 213 euros per month.

Answer:

A

Step-by-step explanation:

I just took the test

9.1 has 2 significant digits so to have 1 you need to round it to a whole number,

in 1 significant digit would be 9