4. Simple Interest: An investment earned 4% simple interest for 8 years. At its maturity, it was worth $5000. What amount was invested? (3​

We conclude that the width of the rectangular garden is 80 feet.

Let's define the variables:

The perimeter of a rectangle of length L and width W is given by the simple formula:

P = 2*(L + W)

The perimeter is equal to 400ft, then:

400ft = 2*(L + W)

And we know that the length is 120ft, then:

L = 120ft.

Replacing the length in the perimeter equation we get:

400ft = 2*(120ft + W)

Now we can solve this linear equation for W.

400ft/2 = 120ft + W

200ft = 120ft + W

200ft - 120ft = W

80ft = W

We conclude that the width of the rectangular garden is 80 feet.

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Answer: 34.3 megabytes

Step-by-step explanation:

To find how many megabytes have been downloaded you must multiply 13.3% by 258. 13.3% can be written as the decimal 0.133

Now let's multiply

[tex]258[/tex]×[tex]0.133[/tex]=34.314

Rounded to the nearest tenth that is 34.3

Answer:  0

Step-by-step explanation:

Step-by-step explanation:

1 meter = 1000 mm

the area of 1.6 m² means the side length of the square is

sqrt(1.6) = 1.264911064... m = 1,264.911064... mm

to create a square out of the rectangular pattern he needs to put e.g. 42 patterns along the 45 mm side and stack 45 patterns on top of the 42 mm side.

the minimum number of needed patterns we get via the last common multiple (LCM) of 42 and 45.

for this we use the prime factorization :

45 ÷ 2 no

45 ÷ 3 = 15

15 ÷ 3 = 5

5 ÷ 3 no

5 ÷ 5 = 1

45 = 3² × 5¹

42 ÷ 2 = 21

21 ÷ 2 no

21 ÷ 3 = 7

7 ÷ 3 no

7 ÷ 5 no

7 ÷ 7 = 1

42 = 2¹ × 3¹ × 7¹

the LCM is the product of the longest streaks of each used prime factor.

LCM(42, 45) = 2¹ × 3² × 5¹ × 7¹ = 2×9×5×7 = 630

(i) he needs 210 patterns to form the smallest square.

this will be

14×45 mm on one side = 630 mm

15×42 mm on the other side = 630 mm

14×15 = 210 patterns.

(ii)

the limit per side length is as established

1,264.911064... mm

starting with the minimum of 630 mm how often can we add 42 mm in one direction and 45 mm in the other, and keep a 15 : 14 ratio between these numbers ?

and we need integer numbers, as we cannot use parts of the patterns (only full patterns).

45 × x <= 1264

x <= 1264/45 = 28.08888888... = integer 28

42 × y <= 1264

y <= 1264/42 = 30.0952381... = integer 30

30/28 = 15/14

so, the ratio is maintained for these numbers.

that means as maximum we can put

30×28 = 840 patterns as square inside the max. allowed area.

the dimensions of this max. square are therefore

30×42 = 1260 mm

28×45 = 1260 mm

the area of this square is then

1260mm × 1260 mm = 1,587,600 mm² = 1.5876 m²

Answer:

0.6 km/hr

Step-by-step explanation:

Speed downstream: 13/5= 2.6 km/hr

Speed upstream: 7/5= 1.4 km/hr

Therefore, Velocity of current is: 1/2 (2.6-1.4) km/hr = 1/2(1.2) km/hr

sorry its a little messy!! not good at drawing with a finger haha. all you need to do is follow pedmas and isolate, knowing that a right angle=90°. hope this helps!<3<3

Answer: [tex]\Large\boxed{x=12}[/tex]

Step-by-step explanation:

Given information

∠1 = x + 42°

∠2 = 3x°

Total Angle = 90° (Right Angle)

Derived formula from the given information

∠1 + ∠2 = Total Angle

Substitute values into the given formula

(x + 42) + (3x) = (90)

Combine like terms

x + 3x + 42 = 90

4x + 42 = 90

Subtract 42 on both sides

4x + 42 - 42 = 90 - 42

4x = 48

Divide 4 on both sides

4x / 4 = 48 / 4

[tex]\Large\boxed{x=12}[/tex]

Hope this helps!! :)

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If [tex]$&f(x)=\frac{19}{x^{2}} \\[/tex] then the inverse function exists  [tex]$&f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].

An inverse function in mathematics exists function which "reverses" the another function.

Let f(x) = y, then the inverse function, [tex]$x=f^{-1}(y)$[/tex]

[tex]$&f(x)=\frac{19}{x^{2}} \\[/tex]

[tex]$&y=\frac{19}{x^{2}} \\[/tex]

[tex]$&x^{2}=\frac{19}{y} \\[/tex]

simplifying the equation, we get

[tex]$&x=\sqrt{\frac{19}{y}} \\[/tex]

[tex]$&x^{-1}=f^{-1}(y)=\sqrt{\frac{19}{y}} \\[/tex]

[tex]$&f^{-1}(y)=\sqrt{\frac{19}{y}},[/tex]  then [tex]$&f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].

If [tex]$&f(x)=\frac{19}{x^{2}} \\[/tex] then the inverse function exists  [tex]$&f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].

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Answer:4 1/12.

Step-by-step explanation: First, you add 3 and a half with a quarter. A half equals two quarters so it equals 3 and 3/4.  Then, you add 3 3/4 with 1/3. A common denominator here is 12. 3/4 times 3/3 = 9/12. 1/3 times 4/4 = 4/12. 9+4 = 13. The answer is more than one so we carry and get 4 and 1/12.

We conclude that the exponential function is:

f(x) = -1*(2)ˣ

Here we know that we have an exponential function of the form:

f(x) = a*b^x

And we know two points on the function, that are:

f(1) = -2 = a*b^1 = a*b

f(2) = -4 = a*b^2

Then we have a system of equations to solve, which is:

-2 = a*b

-4 = a*b^2

From the first equation we can solve:

-2/a = b

Replacing that in the other equation we can get:

-4 = a*(-2/a)^2 = 4/a

a = 4/-4 = -1

Now that we know the value of a, we can get the value of b:

-2/a = b

-2/-1 = 2 = b

In this way, we conclude that the exponential function is:

[tex]f(x) = -1*(2)^x[/tex]

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Answer:

Step-by-step explanation:

The total number of eggs needed for b batches is n = 3b

The total number of eggs needed for 6 batches is 18.

Multiplication is one of the basic mathematical operation that is used to determine the product of two or more numbers. The sign used to denote multiplication is x. Other mathematical operations include addition, subtraction and division.

In order to determine the total number of eggs needed, multiply the number of eggs needed for one batch by the total number of batches.

Total eggs needed = eggs needed for one batch x total number of batches

n = 3 x b

n = 3b

6 x 3 = 18

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Answer: A

Step-by-step explanation:

The domain is the range of possible x that doesn't make y impossible to find, in this case, all real numbers work

The range is the range of possible y that doesn't make x impossible to find in the inverse function, in this case, all real numbers work

The answer is A.

As any number can be replaced in place of x for the function y, the domain of the function is All Real Numbers. Since any number can be inputted, the result will also vary accordingly. Hence, the range of the function is All Real  Numbers.

This is true for the listed function :

[tex]\boxed {y = \sqrt[3]{x-2} - 5}[/tex]

After half of the men go away, the provisions will last for another 16 days.

First, we know that the garrison has provisions for 10 days.

2 days after that, the garrison has provisions for another 8 days. Here we know that the number of men on the garrison is reduced to its half. Then, the time that the provisions in the garrison will last is multiplied by 2.

So we just need to solve the product:

2*8 days = 16 days.

After half of the men go away, the provisions will last for another 16 days.

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Considerando la definición de triángulo rectángulo y sus propiedades, el valor del ángulo que falta de un triángulo rectángulo es 65°.

Un triángulo es un polígono que está determinado por tres segmentos de recta que se denominan lados, o por tres puntos no alineados llamados vértices.

En otras palabras, un triángulo es un polígono conformado por tres lados, así como por tres vértices y tres ángulos interiores.

Una de las propiedades de cualquier triángulo indica que la suma de los ángulos interiores es igual a 180°.

El triángulo rectángulo es una figura geométrica que se considera como un polígono formado por tres lados que forman un ángulo recto y dos ángulos agudos.

En otras palabras, el triángulo rectángulo es aquel que tiene un ángulo interior que es recto, es decir, mide 90º.

En este caso, sabes que:

Entonces, el valor del ángulo faltante se puede calcular mediante:

90° + 25° + ángulo faltante= 180°

Resolviendo:

115° + ángulo faltante= 180°

ángulo faltante= 180° - 115°

ángulo faltante= 65°

Finalmente, el valor del ángulo que falta de un triángulo rectángulo es 65°.

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The number of three-digit positive integers that have three different digits and at least one prime digit are 7960.

The only two components in prime numbers are 1 and the number itself.

Any whole number greater than one is a prime number.

It has exactly two factors—1 and the actual number.

There is just one 2-digit even prime number.

Every pair of prime numbers is always a co-prime.

The product of prime numbers can be used to represent any number.

Three-digit positive integers that have three different digits and at least one prime digit = 3!*4!*10*9 = 7960

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The area of the rectangle with a perimeter of 36 cm is determined as: 80.1  cm².

Area of a rectangle = (length)(width).

Angle of measure 78 degree enclosed by the diagonals of the rectangle

Perimeter = 36 cm

To find the area, we need to find the length and width of the rectangle using the trigonometric ratios.

A side of the rectangle = x cm

The other side = (36 - 2x)/2  = (18 - x) cm

Angle of the diagonal base = (180 - 78)/2 = 51°

Apply the tangent ratio to find x:

Tangent ratio is tan ∅ = opposite/adjacent side of a right triangle. We have the following,

∅ = 51°

Opposite side = x cm

Adjacent side = (18 - x) cm

Tan 51°  =  x/(18 - x)

1.235 = x/(18 - x)

1.235(18 - x) = x

22.23 - 1.235x = x

22.23 = x + 1.235x

22.23 = 2.235x

22.23/2.235 = 2.235x/2.235

x = 9.946 cm (one side length)

The other side length = 18 - x = 18 - 9.946 = 8.054 cm

Area of the rectangle = (9.946)(8.054)

Area of the rectangle = 80.1  cm²

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Answer:

[tex]\lim_{x\rightarrow +\infty } x-1+\frac{1+ln^{2}x}{x} = + \infty[/tex]

[tex]\lim_{x\rightarrow 0 } x-1+\frac{1+ln^{2}x}{x} = + \infty[/tex]

Step-by-step explanation:

[tex]\lim_{x\rightarrow +\infty } x-1+\frac{1+ln^{2}x}{x}[/tex]

[tex]= [\lim_{x\rightarrow +\infty } (x-1)]+[ \lim_{x\rightarrow +\infty } (\frac{1+ln^{2}x}{x})][/tex]

=          +∞             +          0

= +∞

[tex]\lim_{x\rightarrow +\infty } x-1+\frac{1+ln^{2}x}{x}[/tex]

[tex]= [\lim_{x\rightarrow 0 } (x-1)]+[ \lim_{x\rightarrow 0 } (\frac{1+ln^{2}x}{x})][/tex]

=            -1             +      +∞  

= +∞  

Answer:

slope is 4

Step-by-step explanation:

Slope is rise/run

for every 4 units it rises it runs 1 unit

4/1=4

The fraction of the area of polygon H of polygon G's area would be: 1/16.

When a polygon is formed by enlarging or reducing an original polygon by a scale factor, the new polygon formed is similar to the original polygon.

The scale factor = new dimension/original dimension

Given that polygon H is the new polygon formed when polygon G is reduced by a scale factor of 1/4, thus, we would have the following:

Area of polygon H/Area of polygon G = square of the side of polygon H/square of the side of polygon G = square of the scale factor of dilation

Area of polygon H/Area of polygon G = 1²/4² = 1/16

This implies that the area of polygon G is 16 units² while the area of polygon H is 1 units².

Thus, the fraction of the area of polygon H of polygon G's area would be: 1/16.

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Answer:

252

Step-by-step explanation:

using a brute force method in c++

#include <stdio.h>

int main(){

int a, b, c, n = 0;

for(a = 1; a <= 9; a++)

 for(b = 0; b <= 9; b++)

  for(c = 0; c <= 9; c++)

   if((a==3 || b==3 || c==3)) n++;

printf("%d\n", n);

}

Using a trigonometric identity, and considering that the angle is in the fourth quadrant, the tangent of the angle is given as follows:

tan(theta) = -1/4

The following identity is used to relate the measures, considering an angle [tex]\theta[/tex]:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

For this problem, the sine is given as follows:

[tex]\sin{\theta} = -\frac{1}{\sqrt{17}}[/tex]

Then the cosine, which we need to find the tangent, is found as follows:

[tex]\sin^2{\theta} + \cos^2{\theta} = 1[/tex]

[tex]\left(-\frac{1}{\sqrt{17}}\right)^2 + \cos^2{\theta} = 1[/tex]

[tex]\frac{1}{17} + \cos^2{\theta} = 1[/tex]

[tex]\cos^2{\theta} = \frac{16}{17}[/tex]

[tex]\cos{\theta} = \pm \sqrt{\frac{16}{17}}[/tex]

Since the angle is in the fourth quadrant, the cosine is positive, hence:

[tex]\cos{\theta} = \frac{4}{\sqrt{17}}[/tex]

It is the sine of the angle divided by the cosine of the angle, hence:

[tex]\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \frac{-\frac{1}{\sqrt{17}}}{\frac{4}{\sqrt{17}}} = -\frac{1}{4}[/tex]

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Answer: -6

Step-by-step explanation:

[tex]x+3y=-5\\\\3y=-x-5\\\\y=-\frac{1}{3}x-\frac{5}{3}[/tex]

So, since parallel lines have the same slope, the slope of the line we need to find is -1/3.

Substituting into point-slope form, the equation is

[tex]y-2=-\frac{1}{3}(x+7)\\[/tex]

Converting to the required form,

[tex]y-2=-\frac{1}[3}x-\frac{7}{3}\\\\\frac{1}{3}x+y=-\frac{1}{3}\\\\-3x-9y=3[/tex]

So, B-A is equal to -6.

Marketing-oriented firms focus more on what the market needs above every other aspects.

Marketing-oriented firms can be described as firms that major in identifying the needs and wants of customers and then creating specific products that are designed to meet the wants of such customers. One of the important elements of marketing-oriented firms is to ensure there is a demand for whatever products or services they offer.

Some examples of marketing-oriented firms are:

A product-oriented firm focus more resources on and focus on researching and developing quality products rather than focusing more on the market to discover the wants of customers.

Sales-oriented firm tend to focus more on developing its sales force that will promote their products and services.

Marketing-oriented firms stand out and are different from the other two types of firms because they focus more on what the market needs above every other aspects.

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Circle A has center of (6, 7) and a radius of 4, and circle B has a center of (2, 4) and a radius of 16. Now, in order to show that circle A is similar to circle B, we will have to dilate circle A by a scale factor of 4. We can use transformations like dilation and translation to make two circles similar.

Given Information:

For circle A,

Center, C1(x, y) = (6, 7)

Radius, r1 = 4

For circle B,

Center, C2(x, y) = (2, 4)

Radius, r2 = 16

Showing the Two Circles Similar

Now, to make circles A and B similar, we will have to make their centers coincide. It can be done by following the steps listed below:

Hence, we can now see that circle A is similar to circle B.

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Assuming you mean

[tex]\displaystyle \sum_{i=0}^n {}_nC_{i}[/tex]

where

[tex]{}_n C_i = \dbinom ni = \dfrac{n!}{i! (n-i)!}[/tex]

we have by the binomial theorem

[tex]\displaystyle (1 + 1)^n = \sum_{i=0}^n {}_nC_{i} \cdot 1^i \cdot 1^{n-i}[/tex]

so that the given sum has a value of [tex]\boxed{2^n}[/tex].

Answer: [tex]\Large\boxed{2a^3+4ab^2-3a^2b-6b^3}[/tex]

Step-by-step explanation:

Given expression

(2a - 3b) (a² + 2b²)

Apply the FOIL method (First Outer Inner Last)

Please refer to the attachment below for a graphical understanding

= 2a · a² + 2a · 2b² - 3b · a² - 3b · 2b²

Simplify by multiplication

=2 (a · a²) + 4 (a · b²) - 3 (b · a²) - 6 (b · b²)

[tex]\Large\boxed{=2a^3+4ab^2-3a^2b-6b^3}[/tex]

Hope this helps!! :)

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Answer:

Your answer is 2a³ + 4ab² - 3a²b - 6b³.

Step-by-step explanation:

(2a - 3b) (a² + 2b²)

=2a (a² + 2b²) -3b (a² + 2b²)

=2a.a² + 2a.2b² - 3b.a² - 3b.2b²

=2a³ + 4ab² - 3a²b - 6b³        ans.

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The statements which are correct about the equation of circle [tex]x^{2} +y^{2}-2x-8=0[/tex] are 1) The radius of the circle is 3 units,2) The center of the circle lies on the x axis,5) The radius of this circle is the same the radius of the circle whose equation is [tex]x^{2} +y^{2} =9[/tex].

Given the equation of circle be [tex]x^{2} +y^{2}-2x-8=0[/tex].

We are required to find the appropriate statements related to the equation [tex]x^{2} +y^{2}-2x-8=0[/tex].

[tex]x^{2} +y^{2}-2x-8=0[/tex] can be written as under:

[tex]x^{2} +y^{2} -2x+1-9=0[/tex]

[tex]x^{2} +1^{2}-2x+y^{2} -9=0[/tex]

[tex](x-1)^{2}+y^{2}[/tex]-9=0

[tex](x-1)^{2} +y^{2} =9[/tex]

[tex](x-1)^{2}+y^{2} =3^{2}[/tex]

Equation of a circle usually in the form [tex]x^{2} +y^{2} =a^{2}[/tex] in which a is radius.

From the comparison of both the equations we get that radius is 3 units.

From the equation point will be (1,0). It is on the x axis.

Hence the statements which are correct about the equation of circle [tex]x^{2} +y^{2}-2x-8=0[/tex] are 1) The radius of the circle is 3 units,2) The center of the circle lies on the x axis,5) The radius of this circle is the same the radius of the circle whose equation is [tex]x^{2} +y^{2} =9[/tex].

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Answer: 1,2,5 on edge

Step-by-step explanation:
see above ans for work

Answer:

Square Pyramid

Step-by-step explanation:

Thats what a Square Pyramid is