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3 years ago

13

The formula for the volume of a right square pyramid is given below, where a is the side length of the base and h is the height.

Rewrite the formula by solving for a. v= 1/3 a^2h

1 answer:

stira [4]3 years ago

5 0

If it’s v=1/3 *( a^2) * h

Divide both sides by (1/3)*h
v/(1/3*h) = a^2
Or
3v/h = a^2

Square root both sides

Sqrt(3v/h) = a

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The other number is 2 if the given number is 15.

Step-by-step explanation:

One of the numbers is 15, correct?

Let the other number be x.

15 × x = 30

x = 30/15

x = 2.

Let's check:

15 × 2 = 30.

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3 years ago

A cardboard box without a lid is to have a volume of 19,652 cm3. Find the dimensions that minimize the amount of cardboard used.

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Answer:

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.

Step-by-step explanation:

Let the dimension of the cardboard box be x cm by y cm by z cm.

The surface area of the cardboard box without lid is

f(x,y,z)= xy+2xz+2yz.....(1)

Given that the volume of the cardboard is 19,652 cm³.

Therefore xyz =19,652

$\Rightarrow z=\frac{19652}{xy}$ ......(2)

putting the value of z in the equation (1)

$f(x,y)=xy+2x(\frac{19652}{xy})+2y(\frac{19652}{xy})$

$\Rightarrow f(x,y)=xy+\frac{39304}{y})+\frac{39304}{x}$

The partial derivatives are

$f_x=y-\frac{39304}{x^2}$

$f_y=x-\frac{39304}{y^2}$

To find the dimension of the box set the partial derivatives $f_x=0$ and $f_y=0$ .Therefore $y-\frac{39304}{x^2}=0$

$\Rightarrow y=\frac{39304}{x^2}$ .......(3)

and $x-\frac{39304}{y^2}=0$

$\Rightarrow x=\frac{39304}{y^2}$ .......(4)

Now putting the x in equation (3)

$y =\frac {39304}{(\frac{39304}{y^2})^2}$

$\Rightarrow y=\frac{y^4}{39304}$

$\Rightarrow y^3= 39304$

⇒y=34 cm

Then $\Rightarrow x=\frac{39304}{34^2}$ =34 cm.

Putting the value of x and y in the equation (2)

$z=\frac{19652}{34 \times 24}$

=17 cm.

The dimension of the cardboard is 34 cm by 34 cm by 17 cm.

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2 years ago