It says use cofunction but still i didnt understand

There are 9 more pencils then pens in a container. There are 25 writing utensils in the container in all. How many pens are ther

VashaNatasha [74]

In order to determine the number of pens present in the container, we need to set up equations from the given in the problem statement. We do as follows:

let x the number of pencils
y the number of pens

From the statement, 9 more pencils then pens in a container,

x = y + 9

From the statement, there are 25 writing utensils in the container in all,

x + y = 25

We use substitution method to calculate for x and y,

y + 9 + y = 25
2y = 16
y = 8
x = 17

Therefore, there are 8 pens in the container and 17 pencils.

3 0

3 years ago

Rewrite the following integral in spherical coordinates.

lora16 [44]

In cylindrical coordinates, we have $r^2=x^2+y^2$ , so that

$z = \pm \sqrt{2-r^2} = \pm \sqrt{2-x^2-y^2}$

correspond to the upper and lower halves of a sphere with radius $\sqrt2$ . In spherical coordinates, this sphere is $\rho=\sqrt2$ .

$1 \le r \le \sqrt2$ means our region is between two cylinders with radius 1 and $\sqrt2$ . In spherical coordinates, the inner cylinder has equation

$x^2+y^2 = 1 \implies \rho^2\cos^2(\theta) \sin^2(\phi) + \rho^2\sin^2(\theta) \sin^2(\phi) = \rho^2 \sin^2(\phi) = 1 \\\\ \implies \rho^2 = \csc^2(\phi) \\\\ \implies \rho = \csc(\phi)$

This cylinder meets the sphere when

$x^2 + y^2 + z^2 = 1 + z^2 = 2 \implies z^2 = 1 \\\\ \implies \rho^2 \cos^2(\phi) = 1 \\\\ \implies \rho^2 = \sec^2(\phi) \\\\ \implies \rho = \sec(\phi)$

which occurs at

$\csc(\phi) = \sec(\phi) \implies \tan(\phi) = 1 \implies \phi = \dfrac\pi4+n\pi$

where $n\in\Bbb Z$ . Then $\frac\pi4\le\phi\le\frac{3\pi}4$ .

The volume element transforms to

$dx\,dy\,dz = r\,dr\,d\theta\,dz = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi$

Putting everything together, we have

$\displaystyle \int_0^{2\pi} \int_1^{\sqrt2} \int_{-\sqrt{2-r^2}}^{\sqrt{2-r^2}} r \, dz \, dr \, d\theta = \boxed{\int_0^{2\pi} \int_{\pi/4}^{3\pi/4} \int_{\csc(\phi)}^{\sqrt2} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta} = \frac{4\pi}3$

4 0

2 years ago

Suppose an airline policy states that all baggage must be box shaped with a sum of length, width, and height not exceeding 114

NISA [10]

Answer:

Step-by-step explanation:

Represent the length of one side of the base be s and the height by h. Then the volume of the box is V = s^2*h; this is to be maximized.

The constraints are as follows: 2s + h = 114 in. Solving for h, we get 114 - 2s = h.

Substituting 114 - 2s for h in the volume formula, we obtain:

V = s^2*(114 - 2s), or V = 114s^2 - 2s^3, or V = 2*(s^2)(57 - s)

This is to be maximized. To accomplish this, find the first derivative of this formula for V, set the result equal to 0 and solve for s:

dV

----- = 2[(s^2)(-1) + (57 - s)(2s)] = 0 = 2s^2(-1) + 114s - 2s^2

ds

Simplifying this, we get dV/ds = -4s^2 + 114s = 0. Then either s = 28.5 or s = 0.

Then the area of the base is 28.5^2 in^2 and the height is 114 - 2(28.5) = 57 in

and the volume is V = s^2(h) = 46,298.25 in^3

7 0

3 years ago

write a word problem that you can solve by multiplying a mixed number by a whole number. include a solution

Ainat [17]

Jessica is doubling a recipe that calls for 1 1/2 cups of milk. How many cups of milk in total does she need?

2 x 1 1/2 = 3

She needs 3 cups of milk in total.

7 0

3 years ago

Read 2 more answers

Given: PRST square

yuradex [85]

There is no difficulty in this problem until you construct the figures. How can we do it is shown in the attached picture. After drawing PRST, from the point P, we can draw PMKD and later we can complete PMCT as a result. From this picture, we can see that the side of PMCT is also a. Then, the area of this square is $A=a^{2}$

6 0

3 years ago