BIG IDEAS MATH
#14 i
MODELING REAL LIFE You deposit $2000 in an account that earns simple interest at an annual rate of 4%. How long must
you leave the money in the account to earn $500 in interest?


(answer is in years)

The volumes of the cylinder are 565. 56 cubic inches and  1131. 12 cubic inches respectively.

The formula for volume of a cylinder is expressed as;

Volume = πr²h

Where;

Substitute the values

Volume = 3. 142 × 6² × 5

Volume = 3. 142 × 36 × 5

Volume = 565. 56 cubic inches

If the height is doubled, h = 2 × 5 = 10

Substitute the values

Volume = 3. 142 × 6² × 10

Volume = 3. 142 × 36 × 10

Volume = 1131. 12 cubic inches

Thus, the volumes of the cylinder are 565. 56 cubic inches and  1131. 12 cubic inches respectively.

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The fractions represented as power of ten are:

a. 1×(10)⁻³

b. 7× (10)⁻⁵

c. 1 × (10)⁻⁶

d. 5 × (10)⁻⁵

Given :

a. 1/1000

we know that 1/(10)³

when we take the reciprocal, the sign changes  from positive to negative.

that is, 1×(10)⁻³

b. 7/100000

we know that 7/(10)⁵

likewise, we get

that is, 1×(10)⁻⁵

c. 1/10000000

we know that 1/(10)⁷

reciprocate it, we get

1×(10)⁻⁷

d. 3/60000

rearrange it.

3/6×10000

divide it.

1/2×10000

divide and multiply by 10.

10/2×10000×10

multiply it.

5/100000

5 × (10)⁻⁵

Hence we calculated the given fractions as powers of ten.

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Answer:

0.5

Step-by-step explanation:

12/24 = 0.5

6/12 = 0.5

3/6 = 0.5

1.5/3 = 0.5

Answer: 0.5

Answer:

a) 18

b)248

c) 58

Step-by-step explanation:

The formula is:

[tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n-1)d

[tex]a_{n}[/tex]  Is the number at the n term

[tex]a_{1}[/tex] Is the first number in the sequence

n Is the number for the term

d is the common difference.  How much is added each time in the sequence.  In this case, 2.5

Put in the numbers that you know and solve for the unknown.

A:

[tex]a_{8}[/tex] = 0.5 + (8 -1)(2.5)

[tex]a_{8}[/tex]  = 0.5 +(7)(2.5)

[tex]a_{8}[/tex] = 0.5 + 17.5

[tex]a_{8}[/tex] = 18

B:

We follow the same pattern, but we are now looking for [tex]a_{100}[/tex].

[tex]a_{100}[/tex] = 0.5 + (100-1)(2.5)

[tex]a_{100}[/tex] = 0.5 + 247.5

[tex]a_{100}[/tex] = 248

C:

This time we know the value of the number in the sequence, we do not know which term it is.

[tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n - 1)d  Put in what we know

148 = 0.5 + (n -1)2.5  Distribute the 2.5

148 = 0.5 + 2.5n - 2.5  Combine like terms

148 = 3 + 2.5n  Subtract 3 from both sides of the equation

145 = 2.5n  Divide both sides by 2.5

58 = n

Answer : C] 4√6

Thank you !

We get the surface area as 202 inches² and volume as 168 inches³.

The surface area of a cuboid is given by:

Surface area = 2xy + 2xz + 2yz

Here length = x = 7 inches

Breadth = y = 3 inches

Height = z = 8 inches.

Substituting the values in the formula, we get that:

Surface area = A = 2(7)(3) + 2(7)(8) + 2(3)(8) inches²

A = 42 + 112 + 48 inches²

A = 42 + 160 inches²

A = 202 inches²

Volume of a cuboid is given by :

Volume = xyz

Substituting the values, we get that:

Volume = (7) (3) (8) inches³

Volume = 21 × 8 inches³

Volume = 168 inches³.

Therefore, we get the surface area as 202 inches² and volume as 168 inches³.

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Step-by-step explanation:

In this equation, given

f(x) = 3x + 9 and g(x) = 5x -1

So the question is what is the value of f if we put the value of g(x) in f(x)

So to solve the equation we need to replace the value of x in f(x) with g(x)

So it was obtained as below

f(g(x)) = 3(5x-1) + 9

= (15 x - 3) + 9

= 15 x + 6

Answer:

- 2

Step-by-step explanation:

The output of a function is a value of f(x) for a given value for x. From the table, only - 2 matches a value of f(x) and so this is the output.

All the others are values for x so they are inputs to the function

Answer:

Yes.

Step-by-step explanation:

Let's say a=22 and b=-5

According to the Commutative Property of Multiplication, a*b=b*a. Hence, 22*(-5)=(-5)*22.

22*(-5)=

-(22*5)=-110

(-5)*22=

-(5*22)=-110

-110=-110 √

Answer:

no

Step-by-step explanation:

an equation must and always have an equal sign(=) otherwise it is an expression e.g. 2x-3y or x-2

an equation will look like this: a=b or other equations

The larger bottle holds a 700ml volume of water.

What is volume?

Two(2) bottles given in the pic are similar(no difference)

Therefore, the volume/cm height of both(2) bottles will proportional

By using the proportionality rule, [tex]\frac{15}{500} =\frac{21}{x}[/tex]

   x = 700ml(seven hundred)

Hence, the correct answer is 700ml.

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Answer:

The range = 14

Step-by-step explanation:

Definition: (range of a data set)

the range of a data set is the difference between the greatest and smallest  value.

In a box plot :

We should notice the maximum (generally located at the top right) and the minimum value (generally located at the top left)

Then we make the difference.

97.

In our case :

Max value = 24

Min value = 10

Therefore

The range = 24 - 10 = 14

∫dF=F(x,y)=12ax2+bxy+12ℓy2+c=0 is the exact ode.

Specifically, an exact equation is a differential equation that can be solved instantly without the aid of any specialized methods. If the result of a straightforward differentiation, a first-order differential equation (of one variable) is referred to as an exact differential or exact differential. The equation

P(x, y)dy/dx + Q(x, y) = 0,

or in the equivalent alternate notation

P(x, y)dy + Q(x, y)dx = 0,

is exact if

∂P(x, y)/∂x =  ∂Q(x, y)/∂y

In this instance, a function R(x, y) will exist whose partial x-derivative is Q and partial y-derivative is P, and whose equation R(x, y) = c (where c is constant) will implicitly establish a function y that will satisfy the initial differential equation.

An exact ODE is one where the equation can be written as  dF=0  for some  F(x,y). Expanding using chain rule, this means an equation of the form

∂1Fdx+∂2Fdy=0

Now if the second partials of  F  exist and are continuous, they must be equal due to Clairaut. So, given an equation

Mdx+Ndy=0

if we have

∂2M=∂1N

then there is an  F  such that

M=∂1F, N=∂2F

and hence that equation is exact, and the solution is simply

∫dF=0

We have

M=ax+by⇒∂2M=b

N=kx+ℓy⇒∂1N=k

so to be exact, we must have that  b=k .

Then, integrating  M  and  N , we have

F(x,y)=∫Mdx=12ax2+bxy+g(y)

F(x,y)=∫Ndy=bxy+12ℓy2+h(x)

and comparing these two, we can see that

∫dF=F(x,y)=12ax2+bxy+12ℓy2+c=0

is a solution to the equation for any  a,b,c,ℓ . You can solve for  y  by quadratic methods.

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Answer:

Rika

Step-by-step explanation:

The side opposite the 30° angle is longer than the side opposite the 60° angle.

The function is a continuous random variable. The correct option is B.

Continuous random variables are those that have a large number of uncountable possible values. There are countless possible values. In this illustration, there is no limit to the range of possible values for the height of a giraffe in meters, such as 10.5m, 15.22m, 12.0m, etc. There are countless options.

Discrete Variables that have a finite number of possible outcomes are known as random variables. When a dice is rolled, for example, the outcomes could be 1, 2, 3, 4, 5, or 6. There are only six possible values, with nothing in between.

Therefore, the function is a continuous random variable. The correct option is B.

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Answer:

x = 3.2 and No solution.

Step-by-step explanation:

5x - 4 = 12

5x = 12 + 4

5x = 16 (divide both sides by 5 to get x)

5x/5 = 16/5

x = 3.2

–––––––––––––––––––––––––––––

5x-3≥12 AND 12x+5≤-4. Solve it separately.

5x - 4>=12. add 4 on both sides. 5x>=16.

122.500$

First hello I hope you are doing. The first step is to multiply 2.500 with 29 and the result is 72.500$ . Then you add the 50.000$ to 72.500$ which is the raise and you have the result which is 122.500$

I hope I helped you

The equation is 1.95+30+7.20+2/3(m-1.95-30-7.20)+38.50=m

The amount of money  Liza earned is $154.65

It is a method where we find the value of a single unit from the value of multiple units and the value of multiple units from the value of a single unit.

let the money she earn be m.

the remaining money = m-1.95-30-7.20

So, her total earning is,

1.95+30+7.20+2/3(m-1.95-30-7.20)+38.50=m

39.15+38.50+2/3(m-39.15)=m

77.65+2m/3-26.10=m

77.65-26.10=m-2m/3

51.55=(3m-2m)/3

51.55*3=m

m=$154.65

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Answer:

22

Step-by-step explanation:

Simply substitute x = 5 in the function equation

h(5) = 4 (5) + 2 = 20 + 2 = 22

Given ~

[tex] \tt \: h(x) = 4x + 2[/tex]

[tex] \: [/tex]

put the value of x = 5 in given function ~

[tex] \tt \: h(5) = 4(5) + 2[/tex]

[tex] \: [/tex]

[tex] \tt \: h(5) = 20 + 2[/tex]

[tex] \: [/tex]

[tex] \underline{ \boxed{ \tt\red{h(5) = 22}}}{ \green ✓}[/tex]

[tex] \: [/tex]

hope helps ~

Answer:

Step-by-step explanation:

1. a. coordinates of polygon B: (-2, 1), (-4, 1), (-2, 3), (-4, 4)

b. coordinates of polygon C: (-2, -1), (-4, -1), (-2, -3), (-4, -4)

c. coordinates of polygon D: (-2, 1), (-4, 1), (-2, 3), (-4, 4) -> same as polygon B because you after reflecting B to C, they are asking you basically reflect it back to B

2. C (-2, -1)

A linear recurrence relation is a function or sequence in which each term is a linear combination of the terms that came before it.

The recurrence relation exists [tex]$T(n)=\Theta\left(3^n\right)$[/tex].

Recurrence relations are used to simplify complex problems by reducing them to an iterative process based on simpler versions of the problem.

Using the substitution method, we find out that

[tex]T(n) &=n+3 T(n-1) \\[/tex]

[tex]&=n+3(n-1)+3^2 T(n-2) \\[/tex]

[tex]&=n+3(n-1)+3^2(n-2)+3^3 T(n-3) \\[/tex]

[tex]&=\cdots \\[/tex]

[tex]&=n+3(n-1)+3^2(n-2)+\cdots+3^{n-1}(n-(n-1))+3^n T(0) \\[/tex]

simplifying the above equation, we get

[tex]$&=\frac{3^{n+1}-2 n-3}{4}+3^n T(0) \\[/tex]

[tex]&=\Theta\left(3^n\right)[/tex]

Even without doing the full calculation it is not hard to check that [tex]$T(n) \geq 3^{n-1}+3^n T(0)$[/tex], and so [tex]$T(n)=\Omega\left(3^n\right)$[/tex].

A cheap way to obtain the corresponding upper bound is by considering

[tex]$S(n)=T(n) / 3^n$[/tex], which satisfies the recurrence relation  

[tex]$S(n)=S(n-1)+n / 3^n$[/tex].

Repeated substitution then gives

[tex]$\frac{T(n)}{3^n}=\sum_{m=1}^n \frac{m}{3^m}+T(0)[/tex]

Since the infinite series [tex]$\sum_{m=1}^{\infty} \frac{m}{3^m}$[/tex] converges, this implies that [tex]$\frac{T(n)}{3^n}=\Theta(1)$[/tex] and so [tex]$T(n)=\Theta\left(3^n\right)$[/tex]

Therefore, the recurrence relation exists [tex]$T(n)=\Theta\left(3^n\right)$[/tex].

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The length of a rectangle is 27 inches and the width of a rectangle is 12 inches.

Given that, the length of a rectangle is three more than double the width and the perimeter of a rectangle = 78 inches.

We need to find the dimensions of the rectangle.

The perimeter of a rectangle is the total distance of its outer boundary. It is twice the sum of its length and width and it is calculated with the help of the formula: Perimeter = 2(length + width).

Let the width of the rectangle be x.

Then, length=2x+3

Now, perimeter=2(2x+3+x)

⇒ 78=2(3x+3)

⇒ 3x+3= 39

⇒ 3x=36

⇒ x=12 inches

So, width=12 inches and length=2x+3=27 inches

Therefore, the length of a rectangle is 27 inches and the width of a rectangle is 12 inches.

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Answer:

Find the value of t0.05 for a t-distribution with 9 degrees of freedom. round your answer to three decimal places, if necessary.

Step-by-step explanation:

Find the value of t0.05 for a t-distribution with 9 degrees of freedom. round your answer to three decimal places, if necessary.

Answer: -7

Step-by-step explanation: -7 times 2= -14, -14 +9= -5

The Total number of miles that Laney walked is 65 miles.

From the information, Laney walks 13 miles to school from her house and rides the bus home and she walked five days last week.

Therefore, the total number of miles that he walked will be:

= 13 × 5

= 65 miles.

Therefore the miles is 65 miles.

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Work Shown:

[tex]A = (x_1,y_1) = (-2,4) \text{ and } B = (x_2, y_2) = (1,0)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(-2-1)^2 + (4-0)^2}\\\\d = \sqrt{(-3)^2 + (4)^2}\\\\d = \sqrt{9 + 16}\\\\d = \sqrt{25}\\\\d = 5\\\\[/tex]

I used the distance formula.

A slight alternative is to plot the points to form a right triangle. The hypotenuse is the segment from A to B. Then use the pythagorean theorem to find the hypotenuse. Take note how a 3-4-5 right triangle forms.

The common ratio between successive terms in the sequence is 1. 25

Common ratio is defined as the ratio of each term of a geometric sequence to the term before it.

It is also known as the constant factor between consecutive terms of a geometric sequence.

It is  the number multiplied or divided by at each interval.

Given the sequence;

1.5, 1.2, 0.96, 0.768,

The common ration is;

= 1. 5/ 1. 2

= 1. 25

Thus, the common ratio between successive terms in the sequence is 1. 25

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The solution to the expression is x= 18

15 over x=2 equals 3 over 4 can be written as follows

15/x+2 = 3/4

Cross multiply both sides

15×4= 3(x+2)

60= 3x +6

collect the like terms

3x= 60-6

3x= 54

divide both sides by the coefficient of x which is 3

3x/3 = 54/3

x= 18

Hence the value of x in the expression is 18

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