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3 years ago

Please help!! With step by step

Mathematics

1 answer:

Sever21 [200]3 years ago

4 0

Answer:

$R=\$145.12$

Step-by-step explanation:

Compound Interest

This is a well-know problem were we want to calculate the regular payment R needed to pay a principal P in n periods with a known rate of interest i.

The present value PV or the principal can be calculated with

$P=F_a\cdot R$

Solving for R

$\displaystyle R=\frac{P}{F_a}$

Where Fa is computed by

$\displaystyle F_a=\frac{1-(1+i)^{-n}}{i}$

We'll use the provided values but we need to convert them first to monthly payments

$i=7\%=7/(12\cdot 100)=0.00583$

$n=3*12=36$

$\displaystyle F_a=\frac{1-(1+0.00583)^{-36}}{0.00583}$

$F_a=32.386$

Thus, each payment is

$\displaystyle R=\frac{4,700}{32.386}=145.12$

$\boxed{R=\$145.12}$

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Find the maximum and minimum values attained by f(x, y, z) = 5xyz on the unit ball x2 + y2 + z2 ≤ 1.

Allushta [10]

Check for critical points within the unit ball by solving for when the first-order partial derivatives vanish:
$f_x=5yz=0\implies y=0\text{ or }z=0$
$f_y=5xz=0\implies x=0\text{ or }z=0$
$f_z=5xy=0\implies x=0\text{ or }y=0$

Taken together, we find that (0, 0, 0) appears to be the only critical point on $f$ within the ball. At this point, we have $f(0,0,0)=0$ .

Now let's use the method of Lagrange multipliers to look for critical points on the boundary. We have the Lagrangian

$L(x,y,z,\lambda)=5xyz+\lambda(x^2+y^2+z^2-1)$

with partial derivatives (set to 0)

$L_x=5yz+2\lambda x=0$
$L_y=5xz+2\lambda y=0$
$L_z=5xy+2\lambda z=0$
$L_\lambda=x^2+y^2+z^2-1=0$

We then observe that

$xL_x+yL_y+zL_z=0\implies15xyz+2\lambda=0\implies\lambda=-\dfrac{15xyz}2$

So, ignoring the critical point we've already found at (0, 0, 0),

$5yz+2\left(-\dfrac{15xyz}2\right)x=0\implies5yz(1-3x^2)=0\implies x=\pm\dfrac1{\sqrt3}$
$5xz+2\left(-\dfrac{15xyz}2\right)y=0\implies5xz(1-3y^2)=0\implies y=\pm\dfrac1{\sqrt3}$
$5xy+2\left(-\dfrac{15xyz}2\right)z=0\implies5xy(1-3z^2)=0\implies z=\pm\dfrac1{\sqrt3}$

So ultimately, we have 9 critical points - 1 at the origin (0, 0, 0), and 8 at the various combinations of $\left(\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3},\pm\dfrac1{\sqrt3}\right)$ , at which points we get a value of either of $\pm\dfrac5{\sqrt3}$ , with the maximum being the positive value and the minimum being the negative one.

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3 years ago

(a) An angle measures 27°. What is the measure of its complement?

larisa [96]

Answer:

a) 63°

b) 127°

Step-by-step explanation:

Complements are 90° while supplements are 180°.

So you need to find the complement of 27°. So, 90-27=63

To find the supplement, it is 180-53=127

6 0