What is the rate of change for these numbers?

Please help me!!!!!

Bas_tet [7]

Answer: see proof below

Step-by-step explanation:

Use the Sum & Difference Identity: tan (A - B) = (tanA - tanB)/(1 + tanA tanB)

Use the Half-Angle Identity: tan (A/2) = (1 - cosA)/(sinA)

Use the Unit Circle to evaluate tan (π/4) = 1

Use Pythagorean Identity: cos²A + sin²A = 1

Proof LHS → RHS

$\text{Given:}\qquad \qquad \qquad\dfrac{2\tan\bigg(\dfrac{\pi}{4}-\dfrac{A}{2}\bigg)}{1+\tan^2\bigg(\dfrac{\pi}{4}-\dfrac{A}{2}\bigg)}$

$\text{Difference Identity:}\qquad \dfrac{2 \bigg( \frac{\tan\frac{\pi}{4}-\tan\frac{A}{2}}{1+\tan\frac{\pi}{4}\cdot \tan\frac{A}{2}}\bigg)}{1+ \bigg( \frac{\tan\frac{\pi}{4}-\tan\frac{A}{2}}{1+\tan\frac{\pi}{4}\cdot \tan\frac{A}{2}}\bigg)^2}$

$\text{Substitute:}\qquad \qquad \dfrac{2 \bigg( \frac{1-\tan\frac{A}{2}}{1+\tan\frac{A}{2}}\bigg)}{1+ \bigg( \frac{1-\tan\frac{A}{2}}{1+\tan\frac{A}{2}}\bigg)^2}$

$\text{Simplify:}\qquad \qquad \qquad \dfrac{1-\tan^2\frac{A}{2}}{1+\tan^2\frac{A}{2}}$

$\text{Half-Angle Identity:}\qquad \quad \dfrac{1-(\frac{1-\cos A}{\sin A})^2}{1+(\frac{1-\cos A}{\sin A})^2}$

$\text{Simplify:}\qquad \qquad \dfrac{\sin^2 A-1+2\cos A-\cos^2 A}{\sin^2 A+1-2\cos A+\cos^2 A}$

$\text{Pythagorean Identity:}\qquad \qquad \dfrac{1-\cos^2 A-1+2\cos A}{2-2\cos A}$

$\text{Simplify:}\qquad \qquad \qquad \dfrac{2\cos A-2\cos^2 A}{2(1-\cos A)}\\\\.\qquad \qquad \qquad \qquad =\dfrac{2\cos A(1-\cos A)}{2(1-\cos A)}$

= cos A

LHS = RHS: cos A = cos A $\checkmark$

4 0

3 years ago

Megan had some money in her wallet. Her mom gave her $38.25 to go shopping. Now Megan has fifty dollars. How much money did Mega

Ivenika [448]

Answer:

$11.75

Step-by-step explanation:

50-38.25=11.75

6 0

3 years ago

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Jamie has 100 milliliters of a 50% isopropyl alcohol solution. How many milliliters of a 96% isopropyl alcohol solution does she

serious [3.7K]

Answer:

318.2 ml

Step-by-step explanation:

Jamie has 100 ml of a 50% isopropyl alcohol solution.

She wants to prepare a solution of 85% by adding 96% solution of isopropyl alcohol solution.

Let the amount of 96% solution required = x ml

Final amount of 85% isopropyl alcohol solution = (100 + x) ml

Therefore, equation for this situation will be,

100 × (50%) + (x) × (96%) = (100 + x) × (85%)

100(0.5) + 0.96x = 0.85(100 + x)

50 + 0.96x = 85 + 0.85x

0.96x - 0.85x = 85 - 50

0.11x = 35

x = $\frac{35}{0.11}$

x = 318.18 ml

≈ 318.2 ml

Therefore, she will require 318.2 ml of 96% isopropyl alcohol solution.

4 0

3 years ago

Thirteen students entered the business program at Sante Fe College 2 years ago. The following table indicates what each student

gizmo_the_mogwai [7]

Answer:

y=1.003009+0.003453x

or

GPA=1.003009+0.003453(SAT Score)

Step-by-step explanation:

The least square regression equation can be written as

y=a+bx

In the given scenario y is the GPA and x is SAT score because GPA depends on SAT score.

SAT score (X) GPA (Y) X² XY

421 2.93 177241 1233.53

375 2.87 140625 1076.25

585 3.03 342225 1772.55

693 3.42 480249 2370.06

608 3.66 369664 2225.28

392 2.91 153664 1140.72

418 2.12 174724 886.16

484 2.5 234256 1210

725 3.24 525625 2349

506 1.97 256036 996.82

613 2.73 375769 1673.49

706 3.88 498436 2739.28

366 1.58 133956 578.28

sumx=6892

sumy=36.84

sumx²=3862470

sumxy=20251.42

n=13

$b=\frac{(nsumxy)-(sumx)(sumy)}{nsumx^{2}-(sumx)^{2} }$

b=9367.18/2712446

b=0.003453

a=ybar-b(xbar)

ybar=sum(y)/n

ybar=2.833846

xbar=sum(x)/n

xbar=530.1538

a=2.833846-0.003453*(530.1538)

a=1.003009

Thus, required regression equation is

y=1.003009+0.003453x.

The least-squares regression equation that shows the best relationship between GPA and the SAT score is

GPA=1.003009+0.003453(SAT Score)

7 0

3 years ago

Y=-3 - x, how to draw the graph

adell [148]

Answer: See attachment below (graphing)

Step-by-step explanation:

INFORMATION:

The slope-intercept is when y=mx+b

mx=the slope

b=y-intercept

REPHRASE EQUATION:

y=mx+b

y=-3-x

y=(-1)x-3

CONCLUSION:

Slope=-1

Y-intercept=-3

Hope this helps!! :)

Please let me know if you have any questions

6 0

3 years ago

Read 2 more answers