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Answer: There are 2 different unit rates here:

The increase of the tree's height per year

The increase of the tree's trunk per year

Rate of tree's height per year

= 33m / 150 years = 0.22m per year.

Rate of tree's trunk per year

= 3m / 150 years = 0.02m per year.

Answer:

First Blank: “controlled” Second Blank: "a Different" Third Blank: "the same" Last Blank: "Height of the ramp"

Step-by-step explanation:

Answer:

a) 441 gallons

b) -243 fish

Step-by-step explanation:

For a), multiply the 7 gallons of water by the 63 minutes to get 441 total gallons of water added

For b), multiply the 27 fish by 9 days to find the total decrease in fish, which is -243 fish.

Hopefully this helps - let me know if you have any questions!

Answer:

[tex]\displaystyle y' = \frac{\big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \tan^2 x + 5x \big) + \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( 2 \sec^2 x \tan x + 5 \big)}{ \big( \csc^2 x + 3 \big) \big( \sin^2 x + 6 \big)} + \frac{2 \cot x \csc^2 x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big)^2 \big( \sin^2x + 6 \big)} - \frac{2 \cos x \sin x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big) \big( \sin^2 x + 6 \big)^2}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

Derivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]

Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]

Derivative Rule [Basic Power Rule]:

Derivative Rule [Product Rule]:
[tex]\displaystyle (uv)' = u'v + uv'[/tex]

Derivative Rule [Quotient Rule]:
[tex]\displaystyle \bigg( \frac{u}{v} \bigg)' = \frac{vu' - uv'}{v^2}[/tex]

Derivative Rule [Chain Rule]:
[tex]\displaystyle [u(v)]' = u'(v)v'[/tex]

Step-by-step explanation:

*Note:

Since the answering box is way too small for this problem, there will be limited explanation.

Step 1: Define

Identify.

[tex]\displaystyle y = \frac{\cos^2 x - 3\sqrt{x} +6}{\sin^2 x + 6} \times \frac{\tan^2 x + 5x}{\csc^2 x + 3}[/tex]

Step 2: Differentiate

We can differentiate this function with the use of the given derivative rules and properties.

Applying Product Rule:

[tex]\displaystyle y' = \bigg( \frac{\cos^2 x - 3\sqrt{x} + 6}{\sin^2 x + 6} \bigg)' \frac{\tan^2 x + 5x}{\csc^2 x + 3} + \frac{\cos^2 x - 3\sqrt{x} +6}{\sin^2 x + 6} \bigg( \frac{\tan^2 x + 5x}{\csc^2 x + 3} \bigg) '[/tex]

Differentiating the first portion using Quotient Rule:

[tex]\displaystyle \bigg( \frac{\cos^2 x - 3\sqrt{x} + 6}{\sin^2 x + 6} \bigg)' = \frac{\big( \cos^2 x - 3\sqrt{x} + 6 \big)' \big( \sin^2 x + 6 \big) - \big( \sin^2 x + 6 \big)' \big( \cos^2 x - 3\sqrt{x} + 6 \big)}{\big( \sin^2 x + 6 \big)^2}[/tex]

Apply Derivative Rules and Properties, namely the Chain Rule:

[tex]\displaystyle \bigg( \frac{\cos^2 x - 3\sqrt{x} + 6}{\sin^2 x + 6} \bigg)' = \frac{\big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \sin^2 x + 6 \big) - \big( 2 \sin x \cos x \big) \big( \cos^2 x - 3\sqrt{x} + 6 \big)}{\big( \sin^2 x + 6 \big)^2}[/tex]

Differentiating the second portion using Quotient Rule again:

[tex]\displaystyle \bigg( \frac{\tan^2 x + 5x}{\csc^2 x + 3} \bigg) ' = \frac{\big( \tan^2 x + 5x \big)' \big( \csc^2 x + 3 \big) - \big( \csc^2 x + 3 \big)' \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big)^2}[/tex]

Apply Derivative Rules and Properties, namely the Chain Rule again:
[tex]\displaystyle \bigg( \frac{\tan^2 x + 5x}{\csc^2 x + 3} \bigg) ' = \frac{\big( 2 \tan x \sec^2 x + 5 \big) \big( \csc^2 x + 3 \big) - \big( -2 \csc^2 x \cot x \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big)^2}[/tex]

Substitute in derivatives:

[tex]\displaystyle y' = \frac{\big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \sin^2 x + 6 \big) - \big( 2 \sin x \cos x \big) \big( \cos^2 x - 3\sqrt{x} + 6 \big)}{\big( \sin^2 x + 6 \big)^2} \frac{\tan^2 x + 5x}{\csc^2 x + 3} + \frac{\cos^2 x - 3\sqrt{x} +6}{\sin^2 x + 6} \frac{\big( 2 \tan x \sec^2 x + 5 \big) \big( \csc^2 x + 3 \big) - \big( -2 \csc^2 x \cot x \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big)^2}[/tex]

Simplify:

[tex]\displaystyle y' = \frac{\big( \tan^2 x + 5x \big) \bigg[ \big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \sin^2 x + 6 \big) - 2 \sin x \cos x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \bigg]}{\big( \sin^2 x + 6 \big)^2 \big( \csc^2 x + 3 \big)} + \frac{\big( \cos^2 x - 3\sqrt{x} +6 \big) \bigg[ \big( 2 \tan x \sec^2 x + 5 \big) \big( \csc^2 x + 3 \big) + 2 \csc^2 x \cot x \big( \tan^2 x + 5x \big) \bigg] }{\big( \csc^2 x + 3 \big)^2 \big( \sin^2 x + 6 \big)}[/tex]

We can rewrite the differential by factoring and common mathematical properties to obtain our final answer:

[tex]\displaystyle y' = \frac{\big( -2 \cos x \sin x - \frac{3}{2\sqrt{x}} \big) \big( \tan^2 x + 5x \big) + \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( 2 \sec^2 x \tan x + 5 \big)}{ \big( \csc^2 x + 3 \big) \big( \sin^2 x + 6 \big)} + \frac{2 \cot x \csc^2 x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big)^2 \big( \sin^2x + 6 \big)} - \frac{2 \cos x \sin x \big( \cos^2 x - 3\sqrt{x} + 6 \big) \big( \tan^2 x + 5x \big)}{\big( \csc^2 x + 3 \big) \big( \sin^2 x + 6 \big)^2}[/tex]

∴ we have found our derivative.

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Topic: Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Answer:

Below,

Step-by-step explanation:

f(x)=-(x+2)^2+4

(a) Vertex form

(b)  a = -1 , h = -2 and k = 4.

     vertex = (-2, 4)

(c) The value of a, -1 will invert it vertically or we can describe it as a reflection in the line y = 4. The graph will change from a U shape to a ∩ shape.

Hi again! The answer is 200 students

Would you like me to help explain so you can do these questions on your own? :)

Answer:

6.4

Step-by-step explanation:

she has been riding her bike for 6.4 hours.

Answer:Compute the variance: 49+25+16+9+1+16+36+648=27.

The variance is a measure of the dispersion and its value is lower for tightly grouped data than for widely spread data. In the example above, the variance is 27. What does it mean to say that tightly grouped data will have a low variance? You can probably already imagine that the size of the variance also depends on the size of the data itself. Below we see ways that mathematicians have tried to standardize the variance.

Step-by-step explanation:

Answer:

1/[tex]\sqrt{2}[/tex] - 1/4

Step-by-step explanation:

We can write 2[tex]\sqrt{8}[/tex] as 2*[tex]\sqrt{4}[/tex]*[tex]\sqrt{2}[/tex]

That becomes 4[tex]\sqrt{2}[/tex], since the square root of 4 is 2 (±2, to be exact)

(4/(4[tex]\sqrt{2}[/tex])  - ( [tex]\sqrt{2}[/tex]/4[tex]\sqrt{2}[/tex])

1/[tex]\sqrt{2}[/tex] - 1/4

[tex]f(x) =2 x+1 \\\\ \text{If x = 2,}\\\\f(2) = 2(2) +1 = 4+1 = 5[/tex]

Answer:

5

Step-by-step explanation:

plug 2 in for x

f(2) = 2(2) + 1

f(2) = 5 [Asnwer]

PLEASE RATE!! I hope this helps!!

If you have any questions comment below

Answer:

Distributive property. When you factor out a GCF, from 12x + 6y, you would get 6(2x+y).

Step-by-step explanation:

12x + 6y is equivalent to 2x + y using the distributive property of multiplication over addition.

A number system is defined as a system of writing to express numbers.

The distributive property states that a(b + c) = ab + ac, where a, b, and c are any real numbers.

We can apply this property to the expression 2x + y by multiplying 2x by 6 and y by 6, giving:

12x + 6y = 6(2x) + 6(y)

Then, we can use the distributive property again to simplify the expression inside the parentheses:

6(2x) + 6(y) = 6(2x + y)

Substituting this expression back into the original equation, we get:

12x + 6y = 6(2x + y)

Simplifying further by dividing both sides by 6, we obtain:

2x + y = 12x/6 + 6y/6

2x + y = 2x + y

Therefore, we have shown that 12x + 6y is equivalent to 2x + y using the distributive property of multiplication over addition.

To learn more on Number system click:

https://brainly.com/question/22046046

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Answer:

d = 7.2

Step-by-step explanation:

12 + d = -4.8

subtract 12 from both sides and you get

d = 7.2

Answer:

  -16.8

Step-by-step explanation:

The equation can be solved by performing the inverse of the operation that is done to the variable. Here, 12 is added to the variable. The inverse of that operation is addition of the "additive inverse", or "opposite" of 12:

  12 + d = -4.8 . . . . . . given equation

  12 -12 +d = -4.8 -12 . . . . . the same operation is done to both sides

  d = -16.8

The value of d that makes the equation true is -16.8.

_____

The addition (subtraction) property of equality tells you that adding (subtracting) the same number from both sides of an equation will not change the value of the variable.

Answer:

4a because first, do 6 - 2 which is 4 and combine with a.

Please mark brainliest, thanks!

Answer:

27.6

Step-by-step explanation:

1) Say bens depth is x

Cam's depth is x + 4.2

Andrew's depth is x + 4.2 + 16 or x + 11.8

1) andrew's depth = x + 11.8, where x is 15.8.

Therefore, his depth is 27.6 meters.

Answer:

66 (no other answer)

Step-by-step explanation:

The number can be divided by 2 so its even. Its a multiple of 11. It is between 20 and 79. it is a multiple of 3.

Answer:

Step-by-step explanation: 5.6 would be the add on to 27 so since it included the key word '' left'' you'd substract the two and you'd get your answer, just use calculator. You got this!!!

Go sub to clan fbib !!!!!

Answer:

45 divide by 19 =2 r 7 thanks

Step-by-step explanation:

mean = sum of all data points / number of data points

(22+16+18+14+16+34+20)/7 = 140/7 = 20

median is the number, where half of the data points are smaller, and the other half are larger.

first we need to sort the data points

14, 16, 16, 18, 20, 22, 34

median = 18

mode is the data point occurring most often.

mode = 16 (occurring 2 times, while the others occur only once).

range is the difference between the largest and the smallest data point.

range = 34 - 14 = 20

Answer:

11.07~

Step-by-step explanation:

2/7 of 15.5 is 4.42857142857 which we can round to 4.43. We can then just do 15.5-4.43 to get our answer of 11.07.

[tex] \sqrt[3]{ {x}^{5}y } = {x}^{ \frac{5}{3} } {y}^{ \frac{1}{3} } \\ [/tex]

Thus the second option is the correct answer

Answer:

72%

Step-by-step explanation:

We have 10 sudents in class 1 and 40 in the other for a total of 50 students.

The 10 in class 1 had a mean of 80 so we do 80x10=800

The mean of the class of 40 is 70, so we do 40x70=2800

we add both of those together to get 3600 then we divide that by 50 because there are 50 students in total.

3600/50=72.

calculate the sides of the rectangle in scale factor model

Length: 18 * 1/4 = 4.5 feet

Width: 16 * 1/4 = 4 feet

Rectangle area in scale factor model:

Length * Width

4.5 * 4

18 feet²

Answer:

Given dimensions of a rectangular dining table:

If a scale factor of [tex]\dfrac14[/tex] is used, then we need to multiply each measurement by [tex]\dfrac14[/tex]:

[tex]\implies \textsf{width}=18 \times \dfrac14[/tex]

                [tex]=4.5 \textsf{ units}[/tex]

[tex]\implies \textsf{length}=16 \times \dfrac14[/tex]

                 [tex]=4 \textsf{ units}[/tex]

Therefore, the scale model of the dining room will be 4.5 by 4 units.

Answer:

just trust yourself

Step-by-step explanation:

Step-by-step explanation:

let ratio be 4x and 3x

sum : 490$

4x+3x = 490

7x=490

x=490/7

Hence x = 70

first daughter gets = 4x =4 x 70 = 280

second daughter gets =3x = 3x 70 = 210

first daugter gets the larger share

Answer:

the volume should be 9.42

Answer:

-9+x=-20

Step-by-step explanation:

I am not so positive on this one, but if you are not doing inequalities, then this should be correct.

"More than" would insinuate that you are adding. In this class, you would have a number to replace that x, but since no number was given, you place the x in its place. x=-11, because when you "add" -9 with something to get -20, it would have to be -11. Negative numbers can be confusing because when you subtract a negative number by another negative, you end up with a negative. Like in this case. Normally, you would just put -11, so it would look like -9 - 11 = -20. But since this says "more than", unless you are doing inequalities, you add.

If you are doing inequalities, then your answer should be this:

-9 > x = -20

I hope this helps!

-No one