Estimate the limit

A
B
C
D

Answer:

Step-by-step explanation:

z1 = -3

z2 = 5i

z1·z2 = (-3)(5i) = -15i = 0 + (-15)i

Then the real and imaginary parts are a = 0, b = -15.

Answer:

The number b is bigger than the number a by [tex]\dfrac{1}{4}[/tex] of a.

Step-by-step explanation:

1. If the number a is smaller than the number b by 1/5 of b, then

[tex]a+\dfrac{1}{5}b=b.[/tex]

Thus,

[tex]a=b-\dfrac{1}{5}b=\dfrac{4}{5}b,\\ \\b=\dfrac{5}{4}a.[/tex]

2. Consider the expression [tex]b=\dfrac{5}{4}a:[/tex]

[tex]b=\dfrac{5}{4}a=\dfrac{4}{4}a+\dfrac{1}{4}a=a+\dfrac{1}{4}a.[/tex]

This gives you that the number b is bigger than the number a by [tex]\dfrac{1}{4}[/tex] of a.

It is found that A is smaller by 1/5 of b.

To analyze the function or expression to make the function uncomplicated or more coherent is called simplifying and the process is called simplification.

We are given that the number a is smaller than the number b by 1/5 of b.

So if we add a and 1/5 of b, we would have b:

a + 1/5b = b

Solving for a we have:

A = b - 1/5b = 4/5b

A = 4/5b

Solving for b divide both sides by 4/5,

B = 5/4a

Since 4/4 = 1, this means b would be bigger than a by 1/4

Learn more about simplification;

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Answer:

3xy^4+y-2/x

Step-by-step explanation:

12x^3y^4 + 4x^2y -8x

-----------------------------------

4x^2

We can break this fraction into pieces

12x^3y^4      4x^2y        8x

-------------- +    ---------   -  ------------

4x^2                4x^2          4x^2

Taking the first piece

12/4 =3

x^3/x^2 =x

y^4/1 =y^4

3xy^4

Taking the second fractions

4/4=1

x^2/x^2 =1

y=y

y

Taking the third fraction

8/4=2

x/x^2 = 1/x

2/x

Adding them back together

3xy^4+y-2/x

Answer:

(6,8.75)

Step-by-step explanation:

we know that

The distance from the x-axis to the point (7,8.75) is equal to the y-coordinate of the point

so

The distance is 8.75 units

therefore

All ordered pairs that have 8.75 as y coordinate, will be at the same distance from the x axis that the given point

It differs because dilation changes the shape but not the orientation or place the shape is located.

Answer:

I'm assuming the difference is that it changes the size of the original image.

Step-by-step explanation:

Transforming, Rotating, and Reflecting never change the size of the original image but Dilution does. In other words, dilution makes it so that the new image and original image are no longer congruent.

ANSWER

D. Ellipse;

[tex]3{x}^{2} +{y}^{2} + 6x - 6y + 3= 0[/tex]

EXPLANATION

The given equation is

[tex]3 {x}^{2} + {y}^{2} = 9[/tex]

Dividing through by 9 gives

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

This is the equation of an ellipse centered at the origin.

If this ellipse has been translated, so that its center is now at (-1,3), then the equation of the translated ellipse becomes

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

We multiply through by 9 to get,

[tex]3 {(x + 1)}^{2} + {(y - 3)}^{2} = 9[/tex]

Expand to obtain;

[tex]3( {x}^{2} + 2x + 1) + {y}^{2} - 6y + 9 = 9[/tex]

Expand to obtain;

[tex]3{x}^{2} + 6x + 3+ {y}^{2} - 6y + 9 = 9[/tex]

Regroup and equate to zero to obtain;

[tex]3{x}^{2} +{y}^{2} + 6x - 6y + 3= 0[/tex]

In order for it to be a rectangle all four angels have to be 90 so:

13x+34+10x+10=90

23x=46

X=2

Final answer:

To solve for q in the equation k = 4pq², divide both sides of the equation by 4p and take the square root of both sides. Simplify to get q = ±(√k)/(2√p).

Explanation:

To solve for q in the equation k = 4pq², we need to isolate the variable q. Here are the steps:

Divide both sides of the equation by 4p to get q² = k/(4p).

Take the square root of both sides to get q = ±√(k/(4p)).

Simplify the square root to q = ±(√k)/(√(4p)).

Simplify further to q = ±(√k)/(2√p).

Therefore, the correct answer is q = ±(√k)/(2√p). This corresponds to option C.

To find the blank number to be a perfect square, divide the middle number inside parenthesis by 2 and square it.

The middle value is 6.

6/2 = 3

3^2 = 9

The missing number is 9.

The number needed to complete the square in the function f(x) = 4(x² − 6x + ____) + 20 is 9, as (-3)² = 9 leads to forming a perfect square trinomial (x - 3)².

The question asks for the number required to complete the square for the quadratic expression within the function f(x) = 4(x² − 6x + ____) + 20.

To complete the square, we must find a value that, when added to the expression x² − 6x, forms a perfect square trinomial. This involves taking half of the coefficient of the x term, which is -6, and squaring it. The coefficient half is -3, and (-3)² equals 9.

Therefore, the answer for the blank is 9. When substituting into the expression, it transforms to (x - 3)², which is the required perfect square trinomial.

Answer:

A line graph

Step-by-step explanation:

A line graph will be the best option because it can show the exact height based on time as a data point. Connecting each data point will then reveal the trend in how the sapling grows and average growth rate among other information can be found.

Final answer:

A line graph is the best choice for displaying the continuous growth of a sapling tree over a period of several weeks, as it shows trends over time effectively.

Explanation:

The best type of graph for showing the height of a sapling tree over several weeks is a line graph. A line graph is designed to show trends over time and is particularly useful when you want to display changes in a variable continuously, such as the growth of a tree's height. The line graph will clearly depict the gradual increase in height with each passing week, allowing for an easy visual interpretation of the data. Other graphs such as a bar graph, circle graph (or pie chart), or a histogram are not as suitable for representing data over time in the same continuous and clear manner as a line graph.

Answer:

It's the first choice.

Step-by-step explanation:

The common ratio is -6/2 = 18/-6 = -3.

2*(-3)^0 = 2*1 = 2.

2*(-3)^1 = -6

2*(-3)^2 = 18

2*(-3)^3 = -54.

So in summation notation is

∞

∑ 2(-3)^n

n=0

The sum using summation notation is given by:

Summation of two times negative three to the power of n from n equals zero to infinity.

i.e. numerically it is given by:

        [tex]\sum_{n=0}^{\infty} 2(-3)^n[/tex]

The alternating series is given by:

                       [tex]2-6+18-54+........[/tex]

The series could also be written in the form:

[tex]=2+(2\times (-3))+(2\times (-3)\times (-3))+(2\times (-3)\times (-3)\times (-3))+....\\\\i.e.\\\\=2\times (-3)^0+2\times (-3)^1+2\times (-3)^2+2\times (-3)^3+.....\\\\i.e.\\\\=\sum_{n=0}^{\infty} 2(-3)^n[/tex]

Answer:

9.09%

Step-by-step explanation:

You over estimated by 2 feet.  Find out how much 2 feet of 22 is by dividing 2 by 22...

2/22 = 0.090909090909

To convert a decimal to a percent, multiply the decimal by 100%

0.0909090909(100%) = 9.0909090909% or 9.09%

Final answer:

The percent error in the student's estimate of the board's length is approximately 9.09%. The exercise demonstrates the value of precise calculations over rough estimates, although some thoughtful guesses can be quite close.

Explanation:

To find the percent error of the student's estimate, we'll use the following formula:

Percent Error = |(Actual Value - Estimated Value) / Actual Value| × 100%

The actual length of the board is 22 feet, and the estimated length is 24 feet. So we can calculate the percent error as follows:

|(22 - 24) / 22| × 100% = |(-2) / 22| × 100% = (2 / 22) × 100% ≈ 9.09%

The percent error in the student's estimate is approximately 9.09%. This calculation underscores the importance of making careful calculations rather than relying on rough guesstimates, which can lead to significant errors in certain situations. However, it can also show that even guesses made with some thought, like the example of 10 feet versus an actual of 12 feet, can sometimes be surprisingly close.

Answer:

-2.4

Step-by-step explanation:

129x40%=51.6

51.6-54=-2.4

hope that helps!!

Answer:

In the first account was invested [tex]\$3,000[/tex]  at 3%

In the second account was invested [tex]\$4,000[/tex]  at 5%

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Interest Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

in this problem we have

First account

[tex]t=1 years\\ P=\$x\\ r=0.03[/tex]

substitute in the formula above

[tex]I=x(0.03*1)[/tex]

[tex]I=0.03x[/tex]

Second account

[tex]t=1 years\\ P=\$(7,000-x)\\ r=0.05[/tex]

substitute in the formula above

[tex]I=(7,000-x)(0.05*1)[/tex]

[tex]I=350-0.05x[/tex]

Remember that

The interest is equal to [tex]\$290[/tex]

so

Adds the interest of both accounts

[tex]0.03x+350-0.05x=\$290[/tex]

[tex]0.05x-0.03x=350-290[/tex]

[tex]0.02x=60[/tex]

[tex]x=\$3,000[/tex]

therefore

In the first account was invested [tex]\$3,000[/tex]  at 3%

In the second account was invested [tex]\$7,000-\$3,000=\$4,000[/tex]  at 5%

$3000 was invested at 3% and $4000 was invested at 5%.

To determine how much was invested in each account, follow these steps:

1. Define variables:

  - Let [tex]\( x \)[/tex] be the amount invested at 3% interest.

  - Let [tex]\( y \)[/tex] be the amount invested at 5% interest.

2. Set up the equations:

  - The total amount invested is $7000:

    [tex]\[ x + y = 7000 \][/tex]

  - The total interest earned is $290. The interest from each account is [tex]\( 0.03x \)[/tex] and [tex]\( 0.05y \),[/tex] respectively:

   [tex]\[ 0.03x + 0.05y = 290 \][/tex]

3. Solve the system of equations:

  From the first equation:

  [tex]\[ y = 7000 - x \][/tex]

  Substitute [tex]\( y \)[/tex] into the second equation:

  [tex]\[ 0.03x + 0.05(7000 - x) = 290 \][/tex]

  Distribute and simplify:

  [tex]\[ 0.03x + 350 - 0.05x = 290 \][/tex]

  [tex]\[ -0.02x + 350 = 290 \][/tex]

  [tex]\[ -0.02x = 290 - 350 \][/tex]

  [tex]\[ -0.02x = -60 \][/tex]

  [tex]\[ x = \frac{-60}{-0.02} \][/tex]

  [tex]\[ x = 3000 \][/tex]

  Now find [tex]\( y \)[/tex]:

  [tex]\[ y = 7000 - x \][/tex]

 [tex]\[ y = 7000 - 3000 \][/tex]

  [tex]\[ y = 4000 \][/tex]

Answer:

  C.  132

Step-by-step explanation:

We assume all angle measures are in degrees. The vertical angles are equal in measure, so ...

  4x = x +36

  3x = 36

  x = 12

  x +36 = 48

  y = 180 -48 = 132

The value of y is 132.

Answer:

g(-12) = -7

Step-by-step explanation:

Answer: g(-12)=-7

Step-by-step explanation:

g(x)= 3/4x+2

g(-12)= 3/4(-12)+2

g(-12)=-9+2

g(-12)=-7

Answer:

k = -2

Step-by-step explanation:

x     -1   0   2     5

f(x)  2   0  -4  -10

ƒ(x) = kx

Substitute a pair of values for x and ƒ(x)

-10 = k×5

Divide each side by 5

k = -2

The constant of variation k = -2.

Answer:

$3.6

Step-by-step explanation:

6%=.06

60*.06=3.6

Answer:

45 minutes.

Step-by-step explanation:

d = 40 - 1/5m

When d = 31 we have:

31 = 40 - 1/5 m

1/5 m = 40 - 31 = 9

m = 9*5

= 45 minutes answer.

Answer:

21) Yes; 23) Yes

Step-by-step explanation:

If the lengths form a right triangle, the sum of the squares of the two shorter sides should equal the square of the longest side (Pythagoras).

21)

5² +  12² =  13²

25 + 144 = 169

       169 = 169

The lengths form a right triangle.

23)

3² +  4² =  5²

9  + 16 = 25

      25 = 25

The lengths form a right triangle.

do you mean theta not 0

Answer:

Sean's collection was 250 times greater than Jeremy's.

Step-by-step explanation:

1.0 x 10^7 is the same as 10,000,000

4.0 x 10^4 is the same as 40,000

Divide 10,000,000 by 40,000 to get the answer. 10,000,000/40,000 = 250

Answer:

35/12 cups

Step-by-step explanation:

Brett's recipe will make [tex]\( \frac{35}{12} \)[/tex] cups of fruit salad, which is approximately 2.92 cups when rounded to two decimal places.

To find the total number of cups of fruit salad Brett's recipe will make, we add the amounts of each fruit together:

1. Apples: 1 1/2 cups

2. Oranges: 3/4 cup

3. Grapes: 2/3 cup

To add these fractions, we need a common denominator. The least common denominator (LCD) of 2, 4, and 3 is 12.

[tex]1. Apples: \(1 \frac{1}{2} = \frac{3}{2}\) cups[/tex]

[tex]2. Oranges: \(\frac{3}{4}\) cup[/tex]

[tex]3. Grapes: \(\frac{2}{3}\) cup[/tex]

Now, we convert each fraction to have a denominator of 12:

[tex]1. Apples: \(\frac{3}{2} \times \frac{6}{6} = \frac{9}{6}\) cups[/tex]

[tex]2. Oranges: \(\frac{3}{4} \times \frac{3}{3} = \frac{9}{12}\) cups[/tex]

[tex]3. Grapes: \(\frac{2}{3} \times \frac{4}{4} = \frac{8}{12}\) cups[/tex]

Now, we add these amounts:

[tex]\(\frac{9}{6} + \frac{9}{12} + \frac{8}{12} = \frac{18}{12} + \frac{9}{12} + \frac{8}{12} = \frac{35}{12}\) cups[/tex]

So, Brett's recipe will make [tex]\( \frac{35}{12} \)[/tex] cups of fruit salad, which is approximately 2.92 cups when rounded to two decimal places.

Answer:

(0,-5)

Step-by-step explanation:

The vertex form of the equation of a circle is [tex](x-h)^2 + (y-k)^2 = r^2[/tex] where (h,k) is the center of the circle and r is the radius. This means that for the equation [tex]x^2 + (y+5)^2 = 25[/tex] the center is (0,-5).

Answer:

Center:  ( − 5 , 2 )

Radius:  5

Step-by-step explanation:

on edge

Answer:

The speeds are 34 mph for the slower car and 68 mph for the faster car.

Step-by-step explanation:

speed = distance/time

Using s for speed, d for distance, and t for time, we have the equation for speed:

s = d/t

Solve for distance, d, by multiplying both sides by t.

d = st

Now we use the given information.

Speed of slower car: s

Speed of faster car: 2s

Distance traveled by faster car: d

Distance traveled by slower car: d - 204

time traveled by faster car = time traveled by slower car = 6

Distance equation for faster car:

d = st

d = 2s * 6

d = 12s     <---- equation 1

Distance equation for slower car:

d = st

d - 204 = s * 6

d - 204 = 6s

d = 6s + 204    <----- equation 2

Now, using equations 1 and 2, we have a system of two equations in two unknowns.

d = 12s

d = 6s + 204

Since the first equation is already solved for d, we can use the substitution method. Substitute 12s for d in the second equation:

12s = 6s + 204

6s = 204

s = 34

The speed of the slower car is 34 mph.

The speed of the faster car is

2s = 2(34) = 68

The speed of the faster care is 68 mph.

Two cars started moving simultaneously where one was twice as fast as the other. After setting x as the speed of the slower car, the equations showed the slower car traveled at 34 mph and the faster car at 68 mph, based on being 204 miles apart after 6 hours.

Two cars started moving at the same time and direction where one car's speed was twice as fast as the other. After 6 hours, they were 204 miles apart. To solve for the speed of each car, let's set up an equation where the speed of the slower car is x miles per hour and the faster car is 2x miles per hour.

The distance covered by each car after 6 hours would then be:

Since the cars are 204 miles apart after 6 hours, the equation can be set up as:

12x - 6x = 204

So the distance difference is:

6x = 204

Divide both sides by 6 to find the speed of the slower car:

x = 34

Therefore, the slower car travels at 34 mph and the faster car travels at 68 mph (twice the speed of the slow car).

Answer: [tex]y=\frac{2}{5}x-8[/tex]

Step-by-step explanation:

By definition, the equation of the line in  slope-intercept form of is:

[tex]y=mx+b[/tex]

Where m is the slope and b is the y-intercept.

Then, given the slope 2/5 and the point, (-10, -5), you can calculate the value of b by susbtituting and solve for it:

[tex]-10=\frac{2}{5}(-5)+b\\ -10=-2+b\\b=-8[/tex]

Substitute this value and the slope into the equation. THerefore, you obtain:

[tex]y=\frac{2}{5}x-8[/tex]

Answer:   7

Step-by-step explanation:

[tex]\dfrac{\overline{CD}+\overline{A F}}{2}=\overline{BE}\\\\\\\dfrac{(18)+(6x-12)}{2}=2x+10\\\\\\(18)+(6x-12)=2(2x+10)\\\\\\6x +6 = 4x +20\\\\2x+6=20\\\\2x=14\\\\\large \boxed{x=7}[/tex]

Answer:

9,150,625.

Step-by-step explanation:

Any  one of the 55 numbers can be combined with any one of the 55 in the other cylinders.

So the number of different combinations are 55^4

= 9,150,625.

The total number of different combinations that can be made with a four-cylinder combination lock that has 55 numbers on each cylinder is 9,150,625. This is calculated using the multiplication principle of counting.

This question involves the principle of counting or combinatorics in mathematics. Specifically, it relates to the multiplication principle, which says that if event A can occur in m ways, and after it happens, event B can occur in n independent ways, then the total number of ways in which both events can occur is calculated as m times n.

In the case of the four-cylinder combination lock with 55 possible numbers for each cylinder, there are 55 ways to choose a number for the first cylinder. Since repetitions are allowed and each choice is independent, there are also 55 ways to choose a number for the second cylinder, 55 ways for the third cylinder, and 55 ways for the fourth cylinder. Using the multiplication principle, we can find the total number of possible lock combinations by calculating 55 * 55 * 55 * 55 = 9,150,625 possible combinations.

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The area of the triangle with points Q(-1,1),R(2,-3), and S(-1,-3) is equal to 6 square units.

In Mathematics and Geometry, the area of a triangle can be calculated by using the following mathematical equation (formula):

Area of triangle, A = 1/2 × b × h

Where:

By substituting the given vertices into the formula for the area of a triangle with coordinates, we have the following;

[tex]A=\frac{1}{2} \times |x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|\\\\A=\frac{1}{2} \times |-1(-3 + 3) + 2(-3 - 1) + (-1)(1 +3)|\\\\A=\frac{1}{2} \times |0 -8 - 4|\\\\A=\frac{1}{2} \times |-12|[/tex]

Area of triangle = 1/2 × 12

Area of triangle = 6 square units.

Answer:

Step-by-step explanation:

f(x) + n - shift the graph of f(x) n units up

f(x) - n - shift the graph of f(x) n units down

f(x - n) - shift the graph of f(x) n units to the right

f(x + n) - shift the graph of f(x) n units to the left

===================================

Look at the picture.

The graph of F(x) shifted 1 unit to the right and 3 units down.

Therefore the equation of the function G(x) is

[tex]G(x)=(x-1)^2-3[/tex]