Evaluate the expression if x=4 and y=72x(3y-4x)^2

Cot^2x csc^2x + 2 csc^2x –cot^2 = 2

kow [346]

Answer:

x = π/2 + πk

Step-by-step explanation:

cot² x csc² x + 2 csc² x − cot² x = 2

Multiply both sides by sin² x:

cot² x + 2 − cos² x = 2 sin² x

Add cos² x to both sides:

cot² x + 2 = 2 sin² x + cos² x

Pythagorean identity:

cot² x + 2 = sin² x + 1

Subtract 1 from both sides:

cot² x + 1 = sin² x

Pythagorean identity:

csc² x = sin² x

Multiply both sides by sin² x:

1 = sin⁴ x

Take the fourth root:

sin x = ±1

Solve for x:

x = π/2 + 2πk, 3π/2 + 2πk

Which simplifies to:

x = π/2 + πk

4 0

3 years ago

A population of bees is decreasing. The population in a particular region this year is 1,250. After 1 year, it is estimated tha

8_murik_8 [283]

To model this situation, we are going to use the decay formula: $A=Pe^{rt}$
where
$A$ is the final pupolation
$P$ is the initial population
$e$ is the Euler's constant
$r$ is the decay rate
$t$ is the time in years

A. We know for our problem that the initial population is 1,250, so $P=1250$ ; we also know that after a year the population is 1000, so $A=1000$ and $t=1$ . Lets replace those values in our formula to find $r$ :
$A=Pe^{rt}$
$1000=1250e^{r}$
$e^{r}= \frac{1000}{1250}$
$e^{r}= \frac{4}{5}$
$ln(e^{r})=ln( \frac{4}{5} )$
$r=ln( \frac{4}{5} )$
$r=-02231$

Now that we have $r$ , we can write a function to model this scenario:
$A(t)=1250e^{-0.2231t}$ .

B. Here we are going to use a graphing utility to graph the function we derived in the previous point. Please check the attached image.

C.
- The function is decreasing
- The function doe snot have a x-intercept
- The function has a y-intercept at (0,1250)
- Since the function is decaying, it will have a maximum at t=0:
$A(0)=1250e^{(-0.2231)(0)$
$A_{0}=1250e^{0}$
$A_{0}=1250$
- Over the interval [0,10], the function will have a minimum at t=10:
$A(10)=1250e^{(-0.2231)(10)$
$A_{10}=134.28$

D. To find the rate of change of the function over the interval [0,10], we are going to use the formula: $m= \frac{A(0)-A(10)}{10-0}$
where
$m$ is the rate of change
$A(10)$ is the function evaluated at 10
$A(0)$ is the function evaluated at 0
We know from previous calculations that $A(10)=134.28$ and $A(0)=1250$ , so lets replace those values in our formula to find $m$ :
$m= \frac{134.28-1250}{10-0}$
$m= \frac{-1115.72}{10}$
$m=-111.572$
We can conclude that the rate of change of the function over the interval [0,10] is -111.572.

7 0

2 years ago

The area of the Pacific Ocean is 63,780,000 square miles. The Atlantic Ocean is 2.27 cross times 10 to the power of 7 square mil

olganol [36]

Answer:

4.11 * 10⁷ square miles

Step-by-step explanation:

Let P = pacific ocean

A = Atlantic ocean

P = 63,780,000 = 6.38 * 10⁷

Since A is 2.27 * 10⁷ square miles smaller than the pacific ocean,

The area of A = 6.38 * 10⁷ - 2.27 * 10⁷

The area of the atlantic ocean = 4.11 * 10⁷ square miles

4 0

2 years ago

Read 2 more answers

Margret works at a bakery. She is paid $300 every week plus $12 an hour for any

vivado [14]

Answer:

300+12x=$450

Step-by-step explanation:

8 0

2 years ago

Can u answer this mathematics question and show your work

Mnenie [13.5K]

See this solution/explanation (answer is '7 meters'):

1. According to the condition 'y' means the height, and 'x' - the length from the start of the lift hill.

2. The phrase 'height of 343 meters' means y=343.

3. From the another side y=x³. If to substitute 343 instead of 'y': 343=x³, - it is possible to find the value of 'x'.

x=<u>7 [m]</u> - how far from the start of the lift hill...

5 0

2 years ago