Why is cos(pi) and cos (-pi) both equal to -1?

The exponential function is P(t)=60(4)^t that can be expressed in terms of feature of the subsequent form: f ( x ) = a x.

Initial population 'a'=60 rabbits

After 1 year, the rabbit population is 240.

An exponential feature is a mathematical feature of the subsequent form: f ( x ) = a x. wherein x is a variable, and a is a regular known as the bottom of the feature.

Equation of the exponential function.

y=ab^x

Put x=0, y=60

60=ab^0

a=60

Put x=1, y=240

240=60(b)^1

b=4

Thus,

y=60(4)^x

P(t)=60(4)^t

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A negative result does not mean that you do not have the disease. It could be a false negative.

The reasons that could be there for a negative result when there is  actually the presence of the disease

The ELISA may not be sensitive enough to detect very low concentration  of the disease agent, as might occur when one is tested soon after infection before a proper immune response occurs.

As HIV buds from the surface of the host cell, it incorporates some of the host cell Human leukocyte antigen into its envelope. False negatives usually occurs during the window between infection and an antibody response to the virus which is known as seroconversion.

Therefore, if the sample gave a negative result for the disease-causing agent, it does not mean that you do not have the disease

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[tex]\textit{Amount for Exponential Decay using Half-Life} \\\\ A=P\left( \frac{1}{2} \right)^{\frac{t}{h}}\qquad \begin{cases} A=\textit{current amount}\dotfill &\\ P=\textit{initial amount}\dotfill &2.5\\ t=\textit{elapsed time}\\ h=\textit{half-life}\dotfill &13 \end{cases} \\\\\\ ~\hfill {\Large \begin{array}{llll} A=2.5\left( \frac{1}{2} \right)^{\frac{t}{13}} \end{array}} ~\hfill[/tex]

well, let's instead find when will there be 1 gram first off, then after that it's all downhill, so A = 1

[tex]\stackrel{A}{1}=2.5\left( \frac{1}{2} \right)^{\frac{t}{13}}\implies \cfrac{1}{2.5}=\left( \frac{1}{2} \right)^{\frac{t}{13}}\implies \log\left( \cfrac{1}{2.5} \right)=\log\left[ \left( \frac{1}{2} \right)^{\frac{t}{13}} \right][/tex]

[tex]\log\left( \cfrac{1}{2.5} \right)=t\log\left[ \left( \frac{1}{2} \right)^{\frac{1}{13}} \right]\implies \cfrac{\log\left( \frac{1}{2.5} \right)}{\log\left[ \left( \frac{1}{2} \right)^{\frac{1}{13}} \right]}=t\stackrel{\textit{after about 17 days and 5 hours}}{\implies 17.19\approx t ~\hfill }[/tex]

We can shown that the triangles are congruent by a translation of 2 units left and then a reflection over the y-axis.

Congruent triangles are triangles that have the same side lengths.

Transformations that keep  the congruence of a triangle are given as follows:

One vertex of the original triangle is given as follows:

A(8,8).

The equivalent vertex on the reflected and translated triangle is given as follows:

A'(6,-8).

Considering the reflection over the y-axis that we can see from the image, the complete rule is given as follows:

(x,y) -> (x - 2, - y).

Meaning that the translation was two units left.

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Answer:

x = 135

Step-by-step explanation:

the 2 situations are similar with ratios of corresponding sides in proportion , that is

[tex]\frac{x}{54}[/tex] = [tex]\frac{90}{36}[/tex] ( cross- multiply )

36x = 54 × 90 = 4860 ( divide both sides by 36 )

x = 135

The measure of z in the polygon is (a) z = 15

From the question, we have the following parameters that can be used in our computation:

Shape = 9-sided polygon

Parameter to calculate = the measure of z

The sum of the interior angles is calculated using the following angle formula

The sum of the interior angles = ( n − 2 ) × 180 ∘ where n is the number of sides

In this case

n = 9

Substitute the known values in the above equation, so, we have the following representation

The sum of the interior angles = (9 − 2) × 180∘

Evaluate

The sum of the interior angles = 1260∘

So, the internal angle is

Angle = 1260/9

Evaluate

Angle = 140

This means that

5z + 65 = 140

So, we have

5z = 75

Divide by 5

z = 15

Hence, the angle is (a) z = 15

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Answer:

91 cents per ounce

Step-by-step explanation:

Divide 13.59 by 15

Answer: $0.91

Step-by-step explanation:

Take $13.59 divided by 15 = $0.91 per ounce

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The width value if A rectangle has an area of 24 cm² are 8, 48, 0.5, 4, and non-defined.

The measurement that expresses the size of a region on a plane or curved surface is called an area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.

Given:

A rectangle has an area of 24 cm²

The function of the area can be written as,

F(A) = l × b, b = A / l

Put l = 3

then b = 24 / 3 = 8 cm

Put l = 0.5

then b = 24 / 0.5 = 48 cm

Put l = 48

then b = 24 / 48 = 0.5 cm

Put l = -6

then b = 24 / -6 = -4 cm

Put l = 0

then b = 24 / 0  = non defined

Therefore, the width value if A rectangle has an area of 24 cm² are 8, 48, 0.5, 4, and non-defined.

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Answer:

Check below/above

Step-by-step explanation:

A) The graph passes through the point (0,1).

The domain is all real numbers.

The range is y>0.

B) it would be a linear equation that shifts the graph vertically.

C) the slope of an exponential function continues to increase, while the slope of a linear function stays the same.

The percentage discount for the good bought by Nora is 17%.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100. The percentage therefore refers to a component per hundred. Per 100 is what the word percent means. It is represented by %.

In this situation, Nora has a loyalty card good for a discount at her local pharmacy. The item she wants to buy is priced at $28, before discount and tax. After the discount, and before tax, the price is $23.24.

The percentage discount will be:

= Discount / Original value × 100

= (28 - 23.24) / 28 × 100

= 4.76 / 28 × 100

= 17%

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Answer: A, B, C, E

Step-by-step explanation:

Answer: 43 small tanks

Step-by-step explanation:

First, take 900 - 212 = 688 fish in small tanks

Then take 688 divided by 16 fish in each tank = 43

So there are 43 smaller tanks

The proportion of the class that scored higher than the student will be 0.0912.

The z-score is a statistical evaluation of a value's correlation to the mean of a collection of values, expressed in terms of standard deviation.

The z-score is given as

z = (x - μ) / σ

Where μ is the mean, σ is the standard deviation, and x is the sample.

The class normal is 85% and the standard deviation is 3%. Then the student from the class who earned 89% higher will be given as,

z = (0.89 - 0.85) / 0.03

z = 0.04 / 0.03

z = 1.3333

Then the probability is given as,

P(x > 89%) = P(z > 1.3333)

P(x > 89%) = 0.091211

The proportion of the class that scored higher than the student will be 0.0912.

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On solving the provided question, by multiplying the give polynomials, we got to know that -  [tex]( 3x^2 - 5x + 7) X (2x^2 + 6x - 4)[/tex] = [tex]6x^4 + 8x^3 - 28x^2 + 62x - 28[/tex], the coefficient of x3 is 8

An mathematical expression known as a polynomial requires that all of the variables' exponents be integers. Each polynomial variable's exponent must be an integer that is not negative. Although a polynomial is made up of a constant and a variable, it cannot be divided by a variable.

Given polynomials are -

[tex]( 3x^2 - 5x + 7) and (2x^2 + 6x - 4)[/tex]

By multiply the given polynomials, we got -

[tex]( 3x^2 - 5x + 7) X (2x^2 + 6x - 4)[/tex]

=  [tex]6x^4 + 8x^3 - 28x^2 + 62x - 28[/tex]

So, the coefficient of x3 is 8

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Answer:

[tex]\textsf{1)} \quad x=\dfrac{y-b}{a}[/tex]

[tex]\textsf{2)} \quad x=\dfrac{y+b}{a}[/tex]

[tex]\textsf{3)} \quad x=2y-2[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{aligned}&\textsf{Given}: \quad &y & =xa+b\\\\&\textsf{Subtract $b$ from both sides}: \quad & y-b&=xa+b-b \\\\&\textsf{Simplify}: \quad &y-b&=xa \\\\&\textsf{Divide both sides by $a$}: \quad &\dfrac{y-b}{a} &=\dfrac{xa}{a} \\\\&\textsf{Simplify}: \quad &\dfrac{y-b}{a} &=x\\\\&\textsf{Switch sides}: \quad & x&=\dfrac{y-b}{a}\end{aligned}}[/tex]

[tex]\boxed{\begin{aligned}&\textsf{Given}: \quad &y & =xa-b\\\\&\textsf{Add $b$ to both sides}: \quad & y+b&=xa-b+b \\\\&\textsf{Simplify}: \quad &y+b&=xa \\\\&\textsf{Divide both sides by $a$}: \quad &\dfrac{y+b}{a} &=\dfrac{xa}{a} \\\\&\textsf{Simplify}: \quad &\dfrac{y+b}{a} &=x\\\\&\textsf{Switch sides}: \quad & x&=\dfrac{y+b}{a}\end{aligned}}[/tex]

[tex]\boxed{\begin{aligned}&\textsf{Given}: \quad & y&=\dfrac{x+2}{2} \\\\&\textsf{Multiply both sides by $2$}: \quad & 2 \cdot y&=2 \cdot \dfrac{x+2}{2} \\\\&\textsf{Simplify}: \quad & 2y&=x+2 \\\\&\textsf{Subtract $2$ from both sides}: \quad & 2y-2&=x+2-2 \\\\&\textsf{Simplify}: \quad & 2y-2&=x \\\\&\textsf{Switch sides}: \quad &x&=2y-2\end{aligned}}[/tex]

The area is 0.25

What is a tangent line?

The straight line that "just touches" the curve at a particular location is known as the tangent line (or simply tangent) to a plane curve in geometry. It was described by Leibniz as the path connecting two points on a curve that are infinitely near together.  A straight line has a slope of f'(c), where f' is the derivative of f, and is said to be tangent to a curve at a point x = c if it passes through the point (c, f(c)) on the curve. Space curves and curves in n-dimensional Euclidean space have a similar definition.

[tex]$$\begin{aligned}& f^{\prime}(x)=\frac{\mathrm{d}}{\mathrm{d} x}\left[x^3-2 x\right] \\& f^{\prime}(x)=\frac{\mathrm{d}}{\mathrm{d} x}\left[x^3\right]-\frac{\mathrm{d}}{\mathrm{d} x}[2 x] \quad \text { Sum } / \text { Rest rule } \\& f^{\prime}(x)=3 x^{3-1}-2 \quad \text { Power rule } \\& f^{\prime}(x)=3 x^2-2 \\&\end{aligned}$$[/tex]

Now finding tangent for a=-1,

[tex]\begin{aligned}& y=f^{\prime}(a)(x-a)+f(a) \\& y=f^{\prime}(-1)(x+1)+f(-1) \\& y=\left(3(-1)^2-2\right)(x+1)+(-1)^3-2(-1) \\& \mathbf{y}=\mathbf{x}+\mathbf{2}\end{aligned}$$[/tex]

For the first area,

[tex]$$\begin{aligned}& A_1: \int_{-2}^{-\sqrt{2}} x+2 d x=\frac{1}{2} x^2+\left.2 x\right|_{-2} ^{-\sqrt{2}} \\& A_1=\frac{1}{2}(-\sqrt{2})^2+2(-\sqrt{2})-\left(\frac{1}{2}(-2)^2+2(-2)\right) \\& A_1=\mathbf{3}-\mathbf{2} \sqrt{\mathbf{2}} \text { squared units } \\& \mathbf{A}_{\mathbf{1}} \approx \mathbf{0 . 1 7 1 5 7 2 8 7 6} \ldots\end{aligned}$$[/tex]

For the second area,

[tex]$$\begin{aligned}& A_2: \int_{-\sqrt{2}}^{-1} x+2-\left(x^3-2 x\right) d x=-\frac{1}{4} x^4+\frac{3}{2} x^2+\left.2 x\right|_{-\sqrt{2}} ^{-1} \\& A_2=-\frac{1}{4}(-1)^4+\frac{3}{2}(-1)^2+2(-1)-\left(-\frac{1}{4}(-\sqrt{2})^4+\frac{3}{2}(-\sqrt{2})^2+2(-\sqrt{2})\right) \\& A_2=-\frac{\mathbf{1 1}}{\mathbf{4}}+\mathbf{2} \sqrt{\mathbf{2}} \text { squared units } \\& \mathbf{A}_{\mathbf{2}} \approx \mathbf{0 . 0 7 8 4 2 7 1 2 4} \ldots .\end{aligned}$$[/tex]

Hence, required area is

[tex]$$\begin{aligned}& A=A_1+A_2 \\& A=3-2 \sqrt{2}-\frac{11}{4}+2 \sqrt{2} \\& \mathbf{A}=\frac{\mathbf{1}}{\mathbf{4}}=\mathbf{0 . 2 5}\end{aligned}$$[/tex]

The area is 0.25

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The area of the pyramid = 340 sq. inches

The area of a square pyramid= b^2+ 2.b.s

                                                  = 10^2 + 2*10*12

                                                  = 340 sq. inches

We can also find out the are by finding area of each sides

Area of square = 10*10 = 100 sq. inches

Area of triangle= 1/2( 12*10)

                           = 60

Area of four triangles = 4*60

                                    = 240

Total surface area = 240+100

                                = 340 sq. inches

Hence the total surface area is 340 sq. inches

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Answer:

7

Step-by-step explanation:

The wave function that satisfies wave equation is D(x,t) = A cos(kx + ωt) that is option C is correct.

This wave function satisfies the wave equation because it can be rewritten as:

D(x,t) = A cos(kx) cos(ωt) - A sin(kx) sin(ωt)

And the left-hand side of the wave equation is simply the sum of the second derivatives of the two cosine and sine functions with respect to x, while the right-hand side is the product of the second derivative of the cosine function with respect to t and the second derivative of the sine function with respect to t.

For the transverse velocity of a particle at point x, the function is given by:

v(x,t) = -Aω sin(kx) cos(ωt) - Ak cos(kx) sin(ωt)

And for the transverse acceleration, the function is given by:

a(x,t) = -Aω^2 cos(kx) cos(ωt) + Ak^2 sin(kx) sin(ωt)

These functions can be derived by taking the first and second derivatives of the wave function with respect to time.

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By the information provided in the question, we got to know that - there are 2598960 ways.

Combining means selecting all or part of a set of objects, regardless of the order in which the objects are selected. Suppose we have a set of three characters.

[tex]^nC_{r} = \frac{n!}{r!(n - r)!}[/tex]

You are choosing a subset of size 5 from a set of size 52.

[tex]Note that the order you are dealt these 5 cards is irrelevant.[/tex] Thus, order doesn’t matter.

Thus, this is a combination problem. The formula for this is:

[tex]Comb(52,5) = \frac{52!}{5! 13!} = 2598960.[/tex]

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The equation is false (we can remove the dependence with the variable x) that is why the equation has no solutions.

Here we have the following linear equation:

5 - 4x - 7 - 10x = -8x - 4 - 6x - 10

We want to see why we don't have any solutions, to check that, we need to simplify both sides of the equation. Let's start by grouping like terms:

5 - 4x - 7 - 10x = -8x - 4 - 6x - 10

(5 - 7) + (-4x - 10x) = (-4 - 10) + (-8x - 6x)

-2 - 14x = -14 - 14x

Notice that now we can add 14x in both sides of the equation, then we will get:

-2 -14x + 14x = -14 - 14x + 14x

-2 = -14

So we have a false equation, this happens because the equation does not truly depend of the value of x.

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Answer:

8750 Rs

Step-by-step explanation:

Saves

12.5 : 100 = x : 10 000

x = 10 000 * 12.5 / 100

x = 100 * 12.5

x = 1250 Rs

Total he spends

10 000 - 1250 = 8750 Rs

-57 decrease

In this equation.

The value of the function at x=0 is 17.78(4.86)^0 = 17.78(1) = 17.78.

ii. The ratio of output values ​​corresponding to an increment of 1 in input values ​​is (4.86)^1 = 4.86.

iii. A percent change of 1 unit is (4.86 - 1) × 100% = 286% increase.

For the function h(x) = 94.71(0.57)^x:

Thus, the value of the function at x=0 is 94.71(0.57)^0 = 94.71(1) = 94.71.

ii. The ratio of the output value corresponding to an increment of 1 in the input value is (0.57)^2 = 0.57.

iii. Percentage change of 1 unit is (0.57 - 1) × 100% = -57 reduction

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The polynomial of third degree whose zeros and degree are given, for zero of (-2), multiplicity 1 and for zero of (3), multiplicity 2 is [tex]f(x) = x^3 - 10x^2 - 15x + 18[/tex]

A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).

From the question statement, a third degree polynomial has a zero of (-2), multiplicity 1 and another zero of (3), multiplicity 2.

We need to determine the polynomial.

Since, (-2) is a zero of the polynomial, then (x+ 2) will be a factor of the polynomial. for zero of (-2), multiplicity is 1, means that the factor of (x + 2) is only multiplied once.

Similarly, (3) is another zero of the polynomial, then (x - 3) will be a factor of the polynomial.

For zero of (3), multiplicity is 2, this means that the factor of (x - 3) will be multiplied twice

Hence, our required polynomial is [tex]f(x) = x^3 - 10x^2 - 15x + 18[/tex]

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The probability that at least 15 in the sample are among the group of athletes who have been told by coaches to neglect their studies is approximately 0.0998.

Probability can be used to make predictions or decisions in a variety of situations, such as in gambling, finance, and science. In these situations, probabilities can be calculated based on statistical data or by using mathematical models.

To find the probability that at least 15 in the sample are among the group of athletes who have been told by coaches to neglect their studies, we can use the binomial cumulative distribution function. This is given by:

$$P(X \ge 15) = \sum_{k=15}^{50} \binom{50}{k} (0.4)^k (0.6)^{50-k}$$

We can calculate this probability using a calculator or computer, or we can approximate it using the normal distribution. To do this, we can use the continuity correction and compute:

$$P(X \ge 15) \approx P\left(\frac{X-n p}{\sqrt{n p (1-p)}} \ge \frac{15 - 50 \cdot 0.4}{\sqrt{50 \cdot 0.4 \cdot 0.6}}\right) = P(Z \ge 1.28)$$

Where $Z$ is a standard normal random variable. Using a standard normal table or calculator, we find that $P(Z \ge 1.28) \approx 0.0998$.

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The 50% of students who passed the second exam also passed the first exam.

The question is about Conditional Probability. Conditional probability is defined as the measure of the probability of event A given that event B has already occurred. It is denoted by P(A | B). Also, the formula for finding the conditional probability is given below:
P(A | B) = P(A ∩ B)/P(B)         ....(1)
In the given question, We have the following information:
A = Students who passed both exams = 40% = 0.4
B = Students who passed only the second = 80% = 0.8
Now, Let
Probability of Students who passed the second exam and first exam = P (A | B) =?

We can find P(A | B) by putting the values of A and B in equation (1)
By putting values we get:
P (A | B) = [tex]=\frac{40\%}{80\%}[/tex] = [tex]\frac{0.4}{0.8}[/tex]

P (A| B) = 0.5 = 50%

So there are half or 50% of the students passed the second exam and also passed the first exam.

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Answer:

120

Step-by-step explanation:

Let the unknown number be x.

Therefore, if 5.76 is 4.8% of x:

[tex]\boxed{\begin{aligned}4.8\%\; \textsf{of} \; x&=5.76\\\\\implies \dfrac{4.8}{100}x&=5.76\\\\4.8x&=576\\\\x&=\dfrac{576}{4.8}\\\\x&=120 \end{aligned}}[/tex]

So the unknown number is 120.

The bacteria culture has an initial size of 79.7409

Given,

The count in a bacteria culture;

After 10 minutes is 400

After 30 minutes is 2000

Assume that the count grows exponentially;

We have to find the initial size of the culture;

Here,

The growth of population can be calculated using;

P = P₀ (1 + r)^t

Here,

Count of bacteria, P = 400 and 2000

Time period, t = 15 and 30

Initial size =  P₀

So,

400 = P₀ × (1 + r)¹⁰

2000 = P₀ × (1 + r)³⁰

Take P₀ common.

Then,

400/(1 + r)¹⁰ = 2000/(1 + r)³⁰

(1 + r)³⁰/ (1 + r)¹⁰ = 2000/400

(1 + r)¹⁰ = 5

1 + r = 1.175

r = 0.175

Now,

The value of P₀ should be determined from the above equation;

That is,

400 = P₀ × (1 + 0.175)¹⁰

P₀ = 400/ (1 + 0.175)¹⁰

P₀ = 79.7409

Therefore,

The initial size of the bacterial culture is 79.7409

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A linear equation is used to prove the mathematical result that two linear expressions, or a linear expression and a constant, are not equivalent.

The mathematical assertion that two linear expressions, or a linear expression and a constant, are not equal is made via a linear equation.

The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included.

The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

A mathematical equation must take the form y=mx+b in order to be classified as a linear function (in which m is the slope and b is the y-intercept).

Therefore, a linear equation is used to prove the mathematical result that two linear expressions, or a linear expression and a constant, are not equivalent.

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Answer:

Domain:  (-∞, 1) ∪ (1, ∞)

Range:  (-∞, -2) ∪ (2, ∞)

Yes, it's a function.

Step-by-step explanation:

Information

The domain of a function is the set of all possible input values (x-values).

From inspection of the given graph:

Therefore, the domain is all values of x except x = 1:

The range of a function is the set of all possible output values (y-values).

From inspection of the given graph:

Therefore, the range is all values of y except between -2 and 2:

A function is a special type of relationship where each input (x-value) is related to exactly one output (y-value).

From inspection of the graph, as the two points where x = 1 are not included, each value in the range (y-values) corresponds to exactly one value in the domain, and so it is a one-to-one function.