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The exponential decay formula can take one of three forms:

f (x) = abˣ , f (x) = a (1 – r)ˣ , P = P₀[tex]e^{-kt}[/tex]

Where,

a (or) P₀ = Initial amount ,b = decay factor, e = Euler’s constant, r = Rate of decay (for exponential decay), k = constant of proportionality

What is Exponential Decay ?

Exponential decay occurs in mathematical functions when the pace by which changes are occurring are decreasing and must thus reach a limitation, which is the horizontal asymptote of an exponential function.

The amount drops gradually, followed by a quick reduction in the speed of change and increases over time. The exponential decay formula is used to determine the decrease in growth. The exponential decay formula can take one of three forms:

f (x) = abx

f (x) = a (1 – r)x

P = P0 e-k t

Where,

a (or) P0 = Initial amount

b = decay factor

e = Euler’s constant

r = Rate of decay (for exponential decay)

k = constant of proportionality

x (or) t = time intervals (time can be in years, days, (or) months, whatever you are using should be consistent throughout the problem).

In exponential decay, the quantity decreases very rapidly at first, and then more slowly. The rate of change decreases over time. The rate of decay becomes slower as time passes. Hence, the exponential decay graph is denoted as

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B: 1000 + 25 + Water deposit (50) + Phone connection (40) + Elect. connection (60)

1000 + 25 + 200 = 1225

The inverse of the function f(x) = [tex]2x^2 - 4[/tex] is

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here, f(x) = [tex]2x^2 - 4[/tex]

Let f(x) = y

[tex]2x^2 - 4[/tex] = y

[tex]2x^2 = y + 4\\x^2 = \frac{y + 4}{2}\\x = \pm \sqrt{\frac{y + 4}{2}}[/tex]

[tex]f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}[/tex]

First option is correct

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The value of m in the inequality expression given as m + 14 < −125 is m  < -139

In the question, the inequality expression is given as

m+14<−125

Express the inequality properly

So, we have the following representation

m + 14 < −125

To start with, we subtract 14 from both sides

This gives

m + 14 - 14 <−125 - 14

Evaluate the difference

m  < -139

Hence, the solution is m  < -139

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A. Congruence means the same.

B. Angle addition postulate states that two angles side by side sum the measure of the two angles.

C. Linear pair theorem states that linear pair angles add up to 180 degrees.

D. Vertical angles theorem states that vertical angles are congruent.

Congruence in angles simply means the same or equals to.  Two angles are congruent when they are the same.

The angle addition postulate states that the measure of an angle formed by two angles side by side is the sum of the measures of the two angles.

The Linear Pair Theorem states that two angles that form a linear pair are supplementary. They add up to 180 degrees.

The Vertical Angle Theorem says the opposing angles of two intersecting lines must be congruent, or identical in value

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The equation of the perpendicular bisector, in slope-intercept form, is: y = 3/5x + 7/5.

The equation of a line can be expressed:

First, find the slope of the line that is bisected perpendicularly that passes through (1, 2) and (-5, 12):

Slope (m) = change in y / change in x = (12 - 2) / (-5 - 1) = 10/-6

Slope (m) = -5/3

The negative reciprocal or -5/3 is 3/5. Therefore, the slope (m) of the perpendicular bisector would be: 3/5.

To write the equation of the line in point-slope form, substitute m = 3/5 and (a, b) = (1, 2) into y - b = m(x - a):

y - 2 = 3/5(x - 1)

Rewrite in slope-intercept form:

y - 2 = 3/5x - 3/5

y = 3/5x - 3/5 + 2

y = 3/5x + 7/5

Rewrite the equation in standard form:

-3/5x + y = 7/5.

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From the graph it can be calculated that Function rule for g(x) g(x) = |x - 7|

What is graph of a function?

The collection of ordered pairs with the (x,y)(x,y) sign is known as the graph of the a function in mathematics, where f(x)=y. x and f seem to be Cartesian coordinates of pts in two-dimensional space, and in the typical case where they are real numbers, they form a subset of the this plane.

Instead of the pairs "((x,y),z)" and "((x,y),z)" as in the definition above, the graph for functions of two variables, i.e. functions for whom the domain consists of pairs "(x,y)," "(x,y)," is typically the set of ordered triples "(x,y,z)" and "(x,y,z)" where "A surface for a constant real-valued function of the two real variables, this set is indeed a sub - set of three-dimensional space.

Graph of a function f(x) is the collection of points of the form (x, f(x))The given graph is a translation of |x| only,
So value of a is 1
Now since the tip of the graph lies on x- axis, k = 0So the value of g(x) is
g(x) = |x - h|g(7) = 0|7 - h| = 07 - h = 0h = 7
Function rule for g(x) g(x) = = |x - 7|

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Using the compound interest formula, if the account compounds at 4.8% monthly and he withdraws $500 monthly, it will take Steve 33 years, 7 months (403 months) to reduce $100,000 to a balance (future value) of $0.

The future value is the compounded present value, while the present value is the discounted future value.

However, with periodic withdrawals, a present value that compounds at an interest rate into the future can become $0 (future value) after some periods.

The time (period) Steve reduces his inheritance to $0 can be computed using an online finance calculator as follows:

I/Y (Interest per year) = 4.8%

PV (Present Value) = $100,000

PMT (Periodic Payment) = $-500

FV (Future Value) = $0

Results:

N (# of periods) = 403.164

Sum of all periodic payments = $-201,581.83

Total Interest $101,581.83

Thus, after 403 monthly withdrawals, Steve's inheritance will be reduced to a $0 future value.

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The claim that all isosceles triangles are equivalent to one another is untrue.

There are two equal sides and two equal angles in an isosceles triangle.

The term is derived from the Greek words iso (same) and Skelos (leg).

A scalene triangle is one in which none of its sides are equal, as opposed to an equilateral triangle, which has all of its sides equal.

A right triangle with two equal-length legs is called an isosceles right triangle.

The corresponding angles would also be congruent because the lengths of the two legs of the right triangle are equal.

As a result, the two acute angles and two legs of an isosceles right triangle are congruent.

Therefore, the claim that all isosceles triangles are equivalent to one another is untrue.

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The value of x is in the parallelogram is 10

In a parallelogram, the opposite sides are parallel to each other.

The opposite sides of a parallelogram are equal in length.

The measurement of opposite angles of a parallelogram is equal.

Like all other quadrilaterals, the sum of all the angles of a parallelogram is 360°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

The adjacent or the adjoining angles of a parallelogram add up to 180°.

9x +8 and 8x + 2 are adjacent angles of the parallelogram

Mathematically, 9x + 8 + 8x + 2 = 180

17x + 10 = 180

17x + = 180 - 10

17x = 170

x = 170/17

x = 10

In conclusion the value of x is 10

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The length of the line segment will be 4 units. Then the correct option is A.

The line joining the two points is known as a line segment. Let one point be (x, y) and another point be (h, k).

Then the distance between the points will be

D² = (x - h)² + (y - k)²

From the graph, the two points are (2, -2) and (2, 2). Then the distance between the points is calculated as,

D² = (2 - 2)² + (2 + 2)²

Simplify the equation, then we have

D² = (2 - 2)² + (2 + 2)²

D² = (4)²

D = 4 units

The length of the line segment will be 4 units. Then the correct option is A.

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The missing diagram is attached below.

Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.

What is congruent?

Congruent refers to things that are exactly the same size and shape. Even if we flip, turn, or rotate the forms, the shape and size should remain the same.

Here,

we have to prove when can we say that the two triangles are congruent.

SSS, SAS, ASA, AAS, and HL.

These tests describe combinations of congruent sides and/or angles that are used to determine if two triangles are congruent.

Hence, two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.

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Answer:

B) x + 1

Step-by-step explanation:

Consecutive integers mean continuous integers ( 1, 2 , 3 ,....). We will get the next number by adding 1 to the previous number.

Eg: 5 , 6 , 7 ,.....

6 = 5 + 1

7 = 6 + 1

So, if first number is x, then the next number will be x + 1

The difference between 2 parts of congruency is as follows.

What is congruent?

Congruence of triangles: 2 triangles square {measure} aforesaid to be congruent if all 3 corresponding sides square measure equal and every one the 3 corresponding angles square measure equal in measure. These triangles may be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with one another. The image of congruity is’ ≅’

Main Body:

ASA vs AAS:

            ASA stands for “Angle, Side, Angle”,

            AAS means “Angle, Angle, Side”

ASA  ===

two triangles are congruent if they need associate equal aspect contained between corresponding equal angles. If the vertices of 2 triangles are in matched correspondence such 2 angles and therefore the enclosed aspect of 1 triangle are congruent, severally, to the 2 angles and therefore the enclosed aspect of the second triangles, then it satisfies the condition that the triangles are congruent.

AAS===

two angles and an opposite facet. AAS is one amongst the 5 ways that to see if 2 triangles are congruent. It states that if the vertices of 2 triangles are in matched correspondence such 2 angles and therefore the facet opposite to at least one of them in one triangle are congruent to the corresponding angles and therefore the non-included facet of the second triangle, then the triangles are congruent.

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The equation of the demand function is D(x) = 1400√(25-x²) + 11400

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

Marginal demand function, D'(x) = -1400x÷√25-x²

Also, we have

D = 17000, when the value of x = 3

To start with, we need to integrate the marginal demand function, D'(x)

So, we have the following representation

D(x) = 1400√(25-x²) + C

Recall that

D = 17000 at x = 3

So, we have

17000 = 1400√(25-3²) + C

Evaluate

17000 = 5600 + C

Solve for C

C = 17000 - 5600

So, we have

C = 11400

Substitute C = 11400 in D(x) = 1400√(25-x²) + C

D(x) = 1400√(25-x²) + 11400

Hence, the function is D(x) = 1400√(25-x²) + 11400

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Answer:

Step-by-step explanation:

1) 4 1/4 + 5 1/2

= 9 3/4

2) 10 4/5 - 6 1/5

= 4 3/5

3) 3 1/4 - 1 3/4

= 1 1/2

4) 3 2/5 + 7 7/10

= 11 1/10

5 ) 10 3/13 + 2 10/13

= 13

Answer:

1

Step-by-step explanation:

Its for sure 1

The ratio of 240 to 160 is part to part ratio.

What is mean by Ratio?

A ratio indicates how many times one number contain in another number. The ratio of two number is written as x : y, which is equivalent to x/y. Where, x and y are individual amount of two quantities. And, Total quantity gives after combine as x + y.

Given that;

The votes for Tigers outnumbered the votes for Lions by a ratio of 240 to 160.

⇒ 240 : 160.

Now,

We know that,

The ratio of part to part will be two different objects.

And, The ratio of part to whole will be the ratio of object to whole object.

Here, The ratio is given as;

⇒ 240 to 160.

⇒ 240 : 160

And, 240 > 160

So, The ratio is part to part ratio.

Thus, The ratio of 240 to 160 is part to part ratio.

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The incenter of any given triangle s the point of congurrency of the angle bisectors of the triangle

we know that

The incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle's incircle.

therefore

Is the point of concurrency of the angle bisectors of the triangle

Part 2)

we know that

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side

therefore

Is a segment drawn  from a vertex to the midpoint of the opposite side

Therefore, the incenter of any given triangle s the point of congurrency of the angle bisectors of the triangle

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Answer:

3

Step-by-step explanation:

Hope this helps..

To solve complex fractions addition the steps are elaborated below

Basically, a complex fraction is a rational expression that has a fraction in its denominator,  numerator or both, or we can simply state that there is at least one least fraction within the overall fraction.To solve complex fractions addition, the steps are first to convert the  denominators  and numerators  into single fractions and then simplify them, the next step is  to add the fractions with  the like denominators, then Simplify it again, then the step is to   Divide the complex fractions by multiplying the numerator by the reciprocal of the denominator.

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The angles m∠ABC = (3·x - 2)° and m∠ABD formed by the ray [tex]\overrightarrow{BA}[/tex], which is an angle bisector of angle m∠CBD = (5·x + 18)°, indicates that m∠ABD is 64°

An angle bisector is a line, segment or ray that divides an angle into two congruent angles.

The information in the question are;

The ray that bisects ∠CBD = [tex]\overrightarrow{BA}[/tex]

The measure of angle, m∠CBD = (5·x + 18)°

The measure of angle, m∠ABC = (3·x - 2)°

Therefore;

(5·x + 18)° = 2 × (3·x - 2)° (definition of angle formed by an angle bisector)

(5·x + 18)° = (6·x - 4)°

(6·x - 5·x)° = (18 + 4)°

x = 22°

m∠CBD = m∠ABD + m∠ABC  (angle addition postulate)

m∠ABD = m∠ABC (angles formed by angle bisector [tex]\overrightarrow{BA}[/tex])

m∠ABC = (3·x - 2)°

Therefore; m∠ABC = (3 × 22 - 2)° = 64°

m∠ABD = m∠ABC = 64°

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1521=1521 therefore a triangle has sides with lengths of 15 inches, 36 inches, and 39 inches  form a right triangle.

What is Right traingle?

A right triangle is one that has an interior angle of 90 degrees. The hypotenuse, which is also the side of the right triangle that faces the right angle, is its longest side. The height and base make up the two arms of the right angle.

When we enter the values of the triangle's legs (15 and 36) and hypotenuse (39),

assuming it is a right triangle, into the Pythagoras theorem, the values of each side of the equation will be equal;

The Pythagoras Theorem is:

[tex]a^2+b^2=c^2[/tex]

Making substitutions and applying the math results in:

[tex]15^2 + 36^2= 39^2[/tex]

225 + 1296 = 1521

1521 = 1521

Therefore, it is a right traingle.

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The concurrency point for medians of a triangle is known as centroid.

A median is a line drawn from one of the vertices of a triangle that bisects the side opposite to the vertex. Since there are 3 vertices in a triangle, there are also 3 medians in a triangle. A median divides the triangle into 2 equal halves.

The 3 medians of a triangle intersect at a point inside the triangle. This point is known as the centroid of the triangle. It is the center point of a triangle and divides the medians in the ratio of 2 : 1.

Thus, the name of concurrency for medians of a triangle is centroid.

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Logarithmic function convert into the expression as a single logarithm to simplify the function.

The expression as a single logarithm for the is [tex]log_{b} (q^{2} t^{8} )[/tex]

The expression as a single logarithm for the is [tex]log_{b} (q^{2} t^{8} )[/tex]

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The slope of the line, using two points (2, 2) and (-2, -3), is calculated as: 5/4.

Using two points on a line, the slope of a line can be calculated by applying the slope formula below:

Slope (m) = change in y / change in x = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex].

Pick two points on the line in the graph given to find the slope of the line. The two points that are marked on the line are:

(2, 2) = [tex](x_1, y_1)[/tex]

(-2, -3) = [tex](x_2, y_2)[/tex]

Substitute the values into the slope formula:

Slope (m) = (-3 - 2) / (-2 - 2)

Slope (m) = -5/-4 = 5/4

The slope of the line is: 5/4.

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Answer:

It will cost Marley £7.20 to paint the toy box. This is calculated by taking the total surface area of the toy box (180000 cm²) and dividing it by the area covered by each tin of paint (7500 cm²). This equals 24, so Marley will need 24 tins of paint. Twenty-four tins multiplied by the cost of each tin (£8.99) equals £215.76, which is the total cost of the paint.

Step-by-step explanation:

Y = 1x – 2 is the equation for linear regression.

A linear regression line is represented by the equation Y = a + bX, where X is the explanatory variable and Y is the dependent variable. The slope of a line, b, is equal to the intercept of that line, which is the value of y when x = 0.

Based on the value of another variable, linear regression analysis can be used to predict the value of a variable. You want to be able to predict the dependent variable. You create a forecast about the value of the other variable using the independent variable.

A positive linear relationship exists if the slope of the line is positive, meaning that when one rises, the other rises as well.

Y = -2 -1 x

Y= -2 -1x

Y = 1x - 2

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a) The system has infinitely many solutions, which are:

(1 - y, y, -3).

b) The system has no solution.

To obtain the number of solutions of each system, the equations are build from the augmented matrix presented.

For item a, the system is given as follows:

Both the left side of the matrix (3 x 3) and the right side (3 x 1), have the same rank, that is, the same number of non-zero rows, hence the system has infinitely many solutions, which are:

For item b, the system is given as follows:

The last statement is false, hence the system has no solutions. It can also be seem by the different ranks of each matrix, with the 3 x 3 matrix having a rank of 2 and the 3 x 1 matrix having a rank of 3.

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We make a ten in first grade by taking 1 from the next left digit of the number in which we are doing the indicated operation.

According to the question,

We have the following information:

Making a ten in first grade

We consider that in making a ten we take 1 from the next left digit of the number from which we are subtracting any other number. Or it is often said to be taking a carry.

Now, this number 1 is treated as 10 instead of 1 and then 10 is added to the digit to ease subtraction in lower grades.

For example, 24-16 can be performed when we add a ten to 4 by taking 1 from 2. Now, 6 can be easily subtracted from 14.

Hence, we make a ten in first grade by taking 1 from the next left digit of the number in which we are doing the indicated operation.

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Simplify: [tex]5x^4 + 35x^3y -30x^2y^2[/tex]

I used distributive property to mulitply.

ie. [tex]5x^2*x^2[/tex]