What is the truth value for the following conditional statement?
p: true
q: true

p → q

T T → F
F T → T
T T → T
T F → T

1. The growth rate equation has a general form of:

y = A (r)^t

The function is growth when r≥1, and it is a decay when r<1. Therefore:

y=200(0.5)^2t                    --> Decay

y=1/2(2.5)^t/6                  --> Growth

y=(0.65)^t/4                       --> Decay

2. We rewrite the given equation (1/3)^d−5 = 81

Take the log of both sides:

(d – 5) log(1/3) = log 81

d – 5 = log 81 / log(1/3)

d – 5 = - 4

Multiply both sides by negative 1:

- d + 5 = 4

So the answer is D

Answer:

The answer is A. OR   3 5/6

Final answer:

Two functions are inverses if, when composed, they return the original input value (identity function). For example, an exponential function and its inverse, the natural log, undo each other. This can be checked with the equations by rearranging and setting equal to isolate constants or applying the functions one after the other.

Explanation:

How to Determine if Two Functions Are Inverses

To determine if two functions, F and G, are inverses of each other, you can compose the functions and see if the result is the identity function. The composition of two inverse functions will output the original input value, meaning that if you take a value x, apply function F to get F(x), and then apply function G to this result, i.e., G(F(x)), you should get x back if F and G are true inverses. The same must hold true in the reverse order, where G is applied first, followed by F: F(G(x)) = x.

To illustrate this concept, let's consider the exponential function and its inverse function, the natural log.Exponential functions are undone by their inverses, natural logs, according to the relation[tex]eln(x) = ln(ex)[/tex]to see this is with the equations involving a constant A, for instance, these can be rearranged to isolate ln A and set them equal to obtain an identity.

Another familiar context is the square and square root functions. To "undo" a square, you take the square root and vice versa. This "undoing" operation is a key characteristic of inverse functions.

Remember that not all functions have inverses, only those that are one-to-one, meaning for each x there is a unique y and vice versa, have inverses that can be defined over their entire domain.

Final answer:

Tom would need to rent the basketball court 15 times for the gym membership to be more cost-effective than paying the non-member rental fee.

Explanation:

To determine how many times Tom would need to rent the basketball court to make the membership cheaper than being a non-member, we can set up a cost equation and find the breakeven point. Let's assume that Tom rents the court x times. For a non-member, the cost is $11 each time, which results in 11x dollars. For a member, the cost includes the one-time membership fee of $135 plus $2 each time the court is rented, summing to 135 + 2x dollars.

To find the breakeven point, we equate the two costs and solve for x:

11x = 135 + 2x

9x = 135

x = 135 / 9

x = 15

Thus, Tom would need to rent the court 15 times for the membership cost to be cheaper than paying as a non-member.

3 inside angles need to = 180, since it is a right triangle, the bottom left angle = 80 so the other 2 angles need to equal 90 degrees

 so you have:

(8x+2) + (9x+3) = 90

 combine like terms to get:

17x+5 = 90

subtract 5 from both sides:

17x = 85

 divide both sides by 17

x = 85/17 = 5

x = 5

The algebraic expression representing the words "6 less than the product of 3 and a number" is shown as 3x - 6.

An algebraic expression is a combination of terms separated by mathematical operations (+, -, *, /).

A term is a combination of numerals and variables where variables are raised to integral powers.

In the question, we are asked to write the algebraic expression that represents the words "6 less than the product of 3 and a number".

Assuming the number to be x.

We first try to make the terms.

The term that will be formed using the words the product of 3 and a number is 3 * x = 3x.

The term using the word 6 is 6.

Now, to complete the expression, we combine the terms.

The words less than implies the difference. Thus the expression can be written as 3x - 6.

Thus, the algebraic expression representing the words "6 less than the product of 3 and a number" is shown as 3x - 6.

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Answer: [tex]3x+4y=16[/tex]

Step-by-step explanation:

Given equation of a line = [tex]3x + 4y = 15[/tex]

i.e. [tex]4y = 15-3x[/tex]

i.e. [tex]y=\dfrac{15}{4}-\dfrac{3}{4}x[/tex]

i.e. [tex]y=-\dfrac{3}{4}x+\dfrac{15}{4}[/tex]  (1)

It is in intercept form [tex]y=mx+c[/tex]   (2) , where m is slope.

By comparing (1) and (2) the slope of line : [tex]m=-\dfrac{3}{4}[/tex]

Also the parallel lines have same slope .

The equation of a line passing through a point (a,b) and having a slope of 'm' is given by :-

[tex](y-b)= m(x-a)[/tex]

Then the equation of the line that passes through (8,-2) and having slope of [tex]m=-\dfrac{3}{4}[/tex] is given by :-

[tex](y-(-2))=- \dfrac{3}{4}(x-8)[/tex]

[tex]4(y+2)=-3(x-8)[/tex]

[tex]4y+8=-3x+24[/tex]

[tex]3x+4y=24-8[/tex]

[tex]3x+4y=16[/tex]

Hence , the  equation of a line that passes through the point (8, −2) and is parallel to the line whose equation is [tex]3x + 4y = 15[/tex] is [tex]3x+4y=16[/tex]

.

Answer: The number of granola bars in one pack = 8

The number of fruit rolls in one pack = 12

Step-by-step explanation:

Let x be the number of granola bars in each pack and y be the number of fruit rolls in each pack .

Then according to the question we have the following system of equations :

[tex]10x+6y=152.....................(1)\\7x+12y=200.......................(2)[/tex]

Multiply equation (1) by 2 we get

[tex]20x+12y=304......................(3)[/tex]

Subtract equation (2) from equation (3), we get

[tex]13x=104\\\\\Rigtarrow x=8[/tex]

Substitute x=8 in (1), we get

[tex]10(8)+6y=152\\\\\Rightarrow6y=152-80\\\\\Rightarrow\ 6y=72\\\Rightarrow y=12[/tex]

Hence, the number of granola bars in one pack = 8

The number of fruit rolls in one pack = 12

Answer:

3*4*7

Step-by-step explanation:

There is total of 84 combinations

The correct answer is 24.

To find the square root of an improper fraction, simplify the fraction if possible, convert it into perfect squares, or use the properties of exponents. If it is already in simple form like 49/100, directly find the square roots of the numerator and denominator separately. Otherwise, use multiplication to manipulate the fraction into a more manageable form for taking square roots.

To find the square root of an improper fraction, the process involves several steps, similar to finding the square root of any number. First, simplify the fraction if possible. In some cases, this may involve multiplying the numerator and denominator by the same factor to make the radical easier to simplify. For instance, if you have an improper fraction like 49/100, you can easily identify that both the numerator and the denominator are square numbers. So, the square root of 49/100 equals the square root of 49 divided by the square root of 100, which simplifies to 7/10.

When the fraction cannot be simplified easily, a different approach is by converting the numerator and denominator into perfect squares, simplifying the expression within the radical, or using properties of exponents to re-express the square root. For example, if you have a fraction like 1/2, you can multiply both the numerator and denominator by 2 to get 2/4. Then you would rewrite the square root as the square root of 2 over the square root of 4, which simplifies to the square root of 2 over 2.

Answer:

(C) [tex]{\angle}MOP=61^{\circ}[/tex]

Step-by-step explanation:

Given: It is given that angle MON is a straight angle and OP is bisects angle MOQ.

To find: The measure of the angle MOP.

Solution: It is given that angle MON is a straight angle and OP is bisects angle MOQ.

Thus, using the straight line property, we get

[tex]{\angle}MOP+{\angle}POQ+{\angle}QON=180^{\circ}[/tex]

[tex]2{\angle}MOP+{\angle}QON=180^{\circ}[/tex]

Substituting the given values, we get

[tex]2{\angle}MOP+58^{\circ}=180^{\circ}[/tex]

[tex]2{\angle}MOP=122^{\circ}[/tex]

[tex]{\angle}MOP=61^{\circ}[/tex]

Hence, option (C) is correct.

Answer: C - 61°

Step-by-step explanation: edge

It is important for us to simplify rational expressions before multiplying or dividing because it makes our calculations easier.

The rational expression is in its simplest form when there is no common factor between the numerator and the denominator.

A rational number is a number that can be written in the form of p/q where q is not equal to zero.

We have,

Some examples of rational numbers are:

38/44 or 39/63

We can divide this rational number into a decimal number but it is easier for us if we can simplify it in its simplest form.

There is no common factor between numerator and the denominator in its simplest form.

The simplest form of 38/44.

= 38/44

= 2 x 19 / 2 x 22

= 19/22

The simplest form of 39/62 is

= 39/63

= 3 x 13 / 3 x 21

= 13/21

Thus,

It is important for us to simplify rational expressions before multiplying or dividing because it makes our calculations easier.

The rational expression is in its simplest form when there is no common factor between the numerator and the denominator.

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Answer:  The correct option is (B) 3.

Step-by-step explanation:  We are given to find the length of the leg s in the triangle shown in the figure.

From the figure, we note that

the given triangle is a right-angled one, where

length of the hypotenuse, h = √18 units,

length of one leg, p = 3 units

and

length of the other leg, s = ?.

From Pythagoras theorem, we have

[tex]h^2=p^2+s^2\\\\\Rightarrow (\sqrt{18})^2=3^2+s^2\\\\\Rightarrow 18=9+s^2\\\\\Rightarrow s^2=18-9\\\\\Rightarrow s^2=9\\\\\Rightarrow s=3.[/tex]

Thus, the length of the leg s is 3 units.

Option (B) is CORRECT.