The dimensions of the rectangular paint of perimeter 104.09 inches are given as follows:
Height: 28.88 inches.Width: 23.16 inches.How to obtain the perimeter of a rectangle?The perimeter of any polygon, including a rectangle, is given by the sum of all the lengths of the outer edges of the figure.
Hence, for a rectangle of height h and width w, the perimeter is given by twice the sum of these dimensions, hence it is given as follows:
P = 2(h + w).
In this problem, the perimeter has the value given as follows:
P = 104.09 inches.
Hence:
2(h + w) = 104.09
h + w = 104.09/2
h + w = 52.045 inches.
The painting is 5.72 inches taller than it is wide, hence:
h = 5.72 + w.
Then the height is obtained as follows:
5.72 + w + w = 52.045
2w = (52.045 - 5.72)
w = (52.045 - 5.72)/2
w = 23.16 inches.
Then the height of the rectangular painting is given as follows:
h = 5.72 + w = 5.72 + 23.16 = 28.88 inches.
More can be learned about the perimeter of a rectangle at https://brainly.com/question/10489198
#SPJ1
What is the simplified value of the exponential expression 27
1/3
Answer:
3
Step-by-step explanation:
[tex]27^{\frac{1}{3}}[/tex]
1 ) Factor the number [tex]27=3^{3}[/tex]
[tex]=\left(3^3\right)^{\frac{1}{3}}[/tex]
2 ) Simplify...
3
Hope this helps! :)
Graph the equation y= - 7/2 x + 2 on the coordinate plane.
Step-by-step explanation:
your intercepts are:
y-intercept:
y = - 7/2x + 2
y = - 7/20 + 2
y = 2
( 0 , 2 )
x-intercept:
y = - 7/2x + 2
0 = - 7/2x + 2
-2 = - 7/2x
-7/2x = -2
2x = 2/7
x = 4/7
( 4/7 , 0 )
Find a polynomial of degree n that has only the given zero(s). (There are many correct answers.)
x = -8,0, 1; n = 3
Answer: f(x) = x³ + 7x² -8x
Step-by-step explanation:
Zeros means that at that x-value, the graph crosses the x-axis and y = 0. Hence, to meet those criteria, the function would be:
[tex]f(x)=x(x+8)(x-1)[/tex]
At those three x-values, the corresponding y-values would equal 0.
Multiply them out to find the polynomial function:
[tex]f(x)=x(x+8)(x-1)=x(x^{2} -x+8x-8)=x^{3} +7x^{2} -8x[/tex]
Function or not
x y
-10 5
4 -4
-8 6
7 5
-10 5
Answer: function
Step-by-step explanation: for ever input the is one output, even though -10,5 showed up multiple times it had the same input and output. So it’s a function …if I remember correctly.
If the 5 is negative on just one, it’s not a function.
so sorry if it’s wrong, best of luck :)
NO LINKS!! Use the diagram below to answer the following questions. Part 3
Answer:
a) Line r
b) Line q
c) Line s
d) Line q
e) Line p
Step-by-step explanation:
A transversal is a line that crosses two other lines in the same plane at two distinct points.
a) The transversal connecting ∠1 and ∠5 is line r.
Line r is the transversal of lines p and q. ∠1 and ∠5 are corresponding angles as they are in the same position relative to lines p and q crossed by the transversal r.b) The transversal connecting ∠7 and ∠14 is line q.
Line q is the transversal of lines r and s.∠7 and ∠14 are alternate interior angles as they are on the inner side of lines r and s, but on opposite sides of the transversal q.c) The transversal connecting ∠8 and ∠11 is line s.
Line s is the transversal of lines p and q. ∠8 and ∠11 are alternate exterior angles as they are on the outer side of lines p and q, but on opposite sides of the transversal s.d) The transversal connecting ∠6 and ∠15 is line q.
Line q is the transversal of lines r and s. ∠6 and ∠15 are alternate exterior angles as they are on the outer side of lines r and s, but on opposite sides of the transversal q.e) The transversal connecting ∠3 and ∠9 is line p.
Line p is the transversal of lines r and s. ∠3 and ∠9 are consecutive interior angles as they are on the inner side of lines r and s, and are on the same side of the transversal p.Find the distance from point A to XZ . Round your answer to the nearest tenth.
The distance is about (Blank)
units.
Using the distance formula, the distance between A to point Z to the nearest tenth is 4.1 .
How to find the distance using the coordinates?The 2D distance formula gives the shortest distance between two points in a two-dimensional plane.
Using the distance formula,
d = √(y₂ - y₁) + (x₂ - x₁)²
let's use the distance formula, to find the distance between point A and XZ.
Using the coordinates,
A(3, 3) and XZ(4, -1)
Therefore,
x₁ = 3
x₂ = 4
y₁ = 3
y₂ = -1
d = √(4 - 3)² + (-1 - 3)²
d = √1² + (-4)²
d = √1 + 16
d = √17
Therefore,
d = 4.12310562562
d = 4.1
learn more on distance here: https://brainly.com/question/11461046
#SPJ1
If 49% of the people in a community use the emergency room at a hospital in one year, a sample of 11 people use the emergency room. Find the probability of more than 5 people use the emergency room.
The probability of more than 5 people is 0.4729
Find the probability of more than 5 people use the emergency room?Since 49% of the people in a community use the emergency room at a hospital in one year, a sample of 11 people use the emergency room. To find the probability of more than 5 people use the emergency room, we use binomial probability.
What is binomial probability?The binomial probability of x is given by P(X = x) = ⁿCₓpˣ(1 - p)ⁿ⁻ˣ
where
n = number of total trialsp = probability of event andx = number of required trialsSince
n = 11, p = 49% = 0.49 and x = 5,We determine the probability of x ≤ 5.
So, P(X ≤ 5) = ¹¹C₀p⁰(1 - p)¹¹⁻⁰ + ¹¹C₁p¹(1 - p)¹¹⁻¹ + ¹¹C₂p²(1 - p)¹¹⁻² + ¹¹C₃p³(1 - p)¹¹⁻³
+ + ¹¹C₄p²(1 - p)¹¹⁻⁴ + ¹¹C₅p⁵(1 - p)¹¹⁻⁵
= ¹¹C₀(0.49)⁰(1 - 0.49)¹¹ + ¹¹C₁(0.49)¹(1 - 0.49)¹⁰ + ¹¹C₂(0.49)²(1 - 0.49)⁹ + ¹¹C₃(0.49)³(1 - 0.49)⁸
+ ¹¹C₄(0.49)⁴(1 - p)⁷ + ¹¹C₅(0.49)⁵(1 - 0.49)⁶
= (1)(1)(0.51)¹¹ + (11)(0.49)(0.51)¹⁰ + (55)(0.49)²(0.51)⁹ + (165)(0.49)³(0.51)⁸
+ (330)(0.49)⁴(0.51)⁷ + (462)(0.49)⁵(0.51)⁶
= 0.0006071 + (11)(0.49)(0.0011904) + (55)(0.2401)(0.0023341) + (165)(0.117649)(0.0045768)
+ (330)(0.0576480)(0.0089741) + (462)(0.0282475)(0.0175963)
= 0.0006071 + 0.0064163 + 0.0308230 + 0.0888452
+ 0.1707218 + 0.2296378
= 0.5270512
≅ 0.5271
So, P(X ≤ 5) = 0.5271, the probability of more than 5 people is P(X ≥ 5) = 1 - P(X ≤ 5)
= 1 - 0.5271
= 0.4729
The probability is 0.4729
Learn more about binomial probability here:
https://brainly.com/question/27371958
#SPJ1
Dan builds identical model cars and sells them online. He records the number of models he completes every couple of hours. The graph shows Dan’s data. Choose two pairs of points on the line and calculate the slopes between them. Is Dan’s productivity constant? Explain
Yes, Dan productivity is constant and slope is given as 3/2.
What is slope?
The slope of a line is also called its gradient or rate of change. The slope formula is the vertical change in y divided by the horizontal change in x, sometimes called rise over run. The slope formula uses two points, (x1, y1) and (x2, y2), to calculate the change in y over the change in x.
Let we take one pair of point (4,6) and (2,3) and other pair of point is (6,9) and (0,0)
The slope between (4,6) and (2,3) is
m = y2-y1/x2-x1
m = 3-6/2-4
=3/2
The slope between (6,9) and (0,0)
m = 0-9/0-6
=3/2
Hence, both the slopes are equal, therefore his work is constant.
To know more about slope, visit:
https://brainly.com/question/28857060
#SPJ1
a tank is filing with petrol at rate of 35 litre per minute. the tank already had 10 liters in it before the filling began. write an equation in standard form for this situation.
The equation for the situation is written as y = 35x + 10
How to write an equation in standard form for this situationInformation gotten from the question include
a tank is filing with petrol at rate of 35 liter per minute
the tank already had 10 liters in it before the filling began
What is linear function ?A linear function consists of functions where the variables has exponents of 1. The graph of linear functions is a straight line graph and the relationship is expressed in the form.
y = mx + c
Definition of variables to suit the problem to be solved
y = total petrol in the tank in liters
m = rate of filling liter per minute = 35
x = number of liters
c = quantity prior to filling = 10
The equation is written as
y = 35x + 10
Learn more about linear functions at:
https://brainly.com/question/29213643
#SPJ1
The population of a city has increased by 15% since it was last measured. If the current population is 27,600, what was the previous population?
The previous population for the city is 23460.
How to calculate the previous population?Given that the population of a city has increased by 15% since it was last measured, the previous population will be:
= 1 - 15%
= 85%
Therefore, the previous population will be:
= Percentage × New population
= 85% × 27600
= 0.85 × 27600
= 23460
Learn more about percentages on:
brainly.com/question/24304697
#SPJ1
Rewrite, using the distributive property. -3(-4x+5)=
By applying the distributive property of multiplication, this mathematical expression can be rewritten as -3(-4x + 5) = 12x - 15.
What is the distributive property of multiplication?The distributive property of multiplication states that when the sum of two (2) or more addends are multiplied by a particular numerical value, the same result and output would be obtained as when each addend is multiplied respectively by the same numerical value, and the products are added together.
Mathematically, the distributive property of multiplication can be represented by this mathematical expression:
a(b + c) = ab + ac.
By applying the distributive property of multiplication to the given mathematical expression, we have the following:
-3(-4x + 5) = -3(-4x) + [-3(5)]
-3(-4x + 5) = 12x - 15.
Read more on distributive property here; brainly.com/question/24014491
#SPJ1
pls help ill give brainliest
Answer:
(-12, 10)
Step-by-step explanation:
You want to solve by elimination the system of equations ...
2x +4y = 16-2x -3y = -6EliminationThe point of elimination is to make one of the variables disappear from the equation.
Here, we observe that the x-coefficients are opposites in the two equations. When we add them, the result is 0: the x-term disappears. Adding the two equations, we have ...
(2x +4y) +(-2x -3y) = (16) +(-6)
y = 10 . . . . . . . . simplified
Finding the other variableUsing this value for y, we can find x. Substituting into the first equation gives ...
2x +4(10) = 16
2x = 16 -40 = -24 . . . . . subtract 40 and simplify
x = -12 . . . . divide by 2
The ordered pair that is the solution is (x, y) = (-12, 10).
What is the slope of a line perpendicular to the line whose equation is 2x+3y=24 Fully simplify your answer.
Answer:
y = -2/3 x + 8
Step-by-step explanation:
here is the answer in the image above
Some people took part in a game. The frequency shows information about their scores.
Score Frequency
1 - 7 19
8 - 10 16
11 - 15 6
16 - 20 14
21 - 35 15
36 - 50 4
Estimate the mean.
Give your answer rounded to 2 decimal places.
points
The mean of the given scores is 15.43
What is a mean?It is the average value of the set given.
It is calculated as:
Mean = Sum of all the values of the set given / Number of values in the set
We have,
Score Mid-values Frequency Total score
1 - 7 4 19 19 x 4 = 76
8 - 10 9 16 16 x 9 = 144
11 - 15 13 6 13 x 6 = 78
16 - 20 18 14 18 x 14 = 252
21 - 35 28 15 28 x 15 = 420
36 - 50 43 4 43 x 4 = 172
Total 74 1142
Mean = 1142 / 74 = 15.43
Thus,
The mean of the given scores is 15.43
Learn more about mean here:
https://brainly.com/question/23263573
#SPJ1
Function or not
x y
6 4
-8 2
9 -4
-3 8
6 -4
-6 4
Answer: not a function
Step-by-step explanation: one input has more than one output and to be a function it must have one output for every input.
I hope it’s right, best of luck :)
Show how to add the following by adding the opposite: 8-(-3)
Solve the compound inequality 3x+1>2x-3>x-11 enter the exact answer in interval notation
The exact interval notation of given equation is x > - 8.
What is interval notation for compound inequalities?
The numbers that bound a continuous set of real numbers can be used to express the continuous set in interval notation. Intervals have many characteristics in common with ordered pairs when written.
We have,
3x+1 > 2x-3 > x-11
Apply the rule that if m > v > n, then m > v and v > n to the given compound inequality to find the solution.
∴ m = 3x + 1,
v = 2x - 3,
n = x - 11.
Hence, by applying the rule of compound inequality we get,
3x + 1 > 2x - 3 and 2x - 3 > x - 11
3x + 1 > 2x - 3
3x + 1 - 1 > 2x - 3 -1
3x > 2x - 4
3x - 2x > 2x - 4 - 2x
x > - 4
2x - 3 > x - 11
2x - 3 + 3 > x - 11 + 3
2x > x - 8
2x - x > x - 8 - x
x > - 8
The two computations above indicate that the acquired intervals, x > - 4 and x > - 8, will be combined to yield the answer to the given inequality.
To learn more about the interval notation for compound inequalities visit,
https://brainly.com/question/17763094
#SPJ1
Find the unit rate.
16 = 12
The equation has a unit rate of 3/4
How to determine the unit rate?From the question, we have the following parameters
16y = 12x
The above equation is a linear equation
In fact, it is a linear equation that shows a proportional relationship
Note that all linear equations have a constant rate of change
So, the proportional relationship is given as
16y = 12x
Make y the subject
y = 12/16x
This gives
y = 3/4x
As a general rule
A proportional relationship is represented as y = kx
Where k represents the unit rate
So, we have the following representation
y = 3/4x and y = kx
By comparing the equations
k = 3/4
Hence, the unit rate of the equation is 3/4
Read more about unit rate at
https://brainly.com/question/28948763
#SPJ1
Possible question
Find the unit rate.
16y = 12x
hii! could someone help with this?
The projected cash receipts for April and the projected cash receipts for May can be found to be:
April - $187, 500May - $206, 250How to find the projected cash receipts?The projected cash receipts in April would be:
= ( 25% x Projected sales for April ) + ( 75% x Projected sales for March)
= (25% x 210, 000) + ( 75% x 180, 000)
= 52, 500 + 135, 000
= $187, 500
The projected cash receipts for May is then:
= ( 25% x Projected sales for May ) + ( 75% x Projected sales for April )
= ( 25% x 195, 000) + ( 75% x 210, 000)
= $ 48, 750 + 157, 500
= $206, 250
Find out more on projected cash receipts at https://brainly.com/question/14821235
#SPJ1
The 3rd term of a GP is 18 and the 7th term is 3⅝, find the GP
The geometric sequence in which the 3rd term is of 18 and the 7th term is of 29/8 is defined as follows:
[tex]a_n = 40(0.67)^{n - 1}[/tex]
What is a geometric sequence?A geometric sequence is a sequence in which the result of the division of consecutive terms yields always the same result, called common ratio q.
The nth term of a geometric sequence is given by the rule presented as follows:
[tex]a_n = a_1q^{n-1}[/tex]
In which [tex]a_1[/tex] is the first term.
The rule for the nth term is also called the general rule.
The general takes the first term as the reference term, however, the 7th term can be written as a function of the 3rd term, as follows:
[tex]a_7 = a_3q^{4}[/tex]
The fraction that represents the mixed number 3 and 5/8 is:
(3 x 8 + 5)/8 = 29/8.
Then the common ratio of the sequence is calculated as follows:
[tex]\frac{29}{8} = 18q^{4}[/tex]
[tex]q^4 = \frac{29}{144}[/tex]
[tex]q = \sqrt[4]{\frac{29}{144}}[/tex]
[tex]q = \left(\frac{29}{144}\right)^{\frac{1}{4}}[/tex]
q = 0.67.
Then the first term of the sequence is obtained as follows:
[tex]a_3 = a_1q^2[/tex]
[tex]a_1 = \frac{a_3}{q^2}[/tex]
[tex]a_1 = \frac{18}{(0.67)^2}[/tex]
[tex]a_1 = 40[/tex]
Then the definition of the geometric sequence is:
[tex]a_n = 40(0.67)^{n - 1}[/tex]
More can be learned about geometric sequences at https://brainly.com/question/24643676
#SPJ1
UpEx delivers packages which cost $1.50 per package to deliver. The fixed cost to run the delivery truck is $198 per day. If the company charges $3.50 per package, how many packages must be delivered daily to make a profit of $32?
The number of packages that need to be delivered daily by UpEx to make a profit of $32 is 115 packages
How to find the number of packages?First, find the contribution margin of each package:
= Charge to deliver package - Cost to deliver package
= 3.50 - 1.50
= $2.00
To find the number of packages needed to make a profit of $32, the formula is:
= (Fixed cost + projected profit) / Contribution margin
= ( 198 + 32) / 2
= 230 / 2
= 115 packages
How to find the quantity to make profit at https://brainly.com/question/17929256
#SPJ1
Please help!!
Will give 5 points asap
The probabilities are given as follows:
a) P(both marbles are red) = 0.167 = 16.7%.
b) P(red then green) = 0.111 = 11.1%.
c) P(green then blue) = 0.083 = 8.3%.
d) P(blue and red, any order) = 0.333 = 33.3%.
How to obtain a probability?The probability of an event in an experiment is calculated as the number of desired outcomes in the context of the experiment divided by the number of total outcomes in the context of the experiment.
In this problem, the are nine marbles, of which four are red, hence the probability of both red is:
p = 4/9 x 3/8 = 0.167 = 16.7%.
(the marble is not replaced, hence for the second marble there will be eight possible marbles, of which 3 are red).
For red then green, the probability is:
p = 4/9 x 2/8 = 0.111 = 11.1%.
For green then blue, the probability is:
p = 2/9 x 3/8 = 0.083 = 8.3%.
For blue or red, there are two possible outcomes, blue then red or red then blue, hence:
p = 3/9 x 4/8 + 4/9 x 3/8 = 0.333 = 33.3%.
More can be learned about probabilities at https://brainly.com/question/14398287
#SPJ1
How many tons are in 12,000 pound
Answer: 12,000pd=6 tons
Step-by-step explanation:
Question 7(Multiple Choice Worth 1 points)
(02.02 MC)
Given the function f(x) = -2x² + 3x + 10, find f(1) and f(3). Choose the statement that is true concerning these two values.
The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10
Given that,
The function is f(x)= -2x² + 3x + 10
We have to find f(1) and f(3).
What is function?The core concept of mathematics' calculus is functions. The functions are special types of relations. In mathematics, a function is represented as a rule that produces a distinct result for each input x.
Take the function
f(x)= -2x² + 3x + 10
Take x as 1
f(1)= -2(1)² + 3(1) +10
f(1)= -2+3+10
f(1) = 11
Now, Take x as 3
f(3)= -2(3)² + 3(3) +10
f(3)= -18+9+10
f(3) = 1
Therefore, The value of the function f(1) and f(3) is 11 and 1 when the function is f(x)= -2x² + 3x + 10
To learn more about function visit: https://brainly.com/question/21145944
#SPJ1
What is the domain of the function shown in the graph below?
{x|x>-4}
{x|x ≥-4}
{yly ≥ 2}
The domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is
[ -4 , ∞ ).
Given, a graph of the function,
{ x | x ≥ -4 }
{ y l y ≥ 2 }
Now, we have to find the domain of the function
As, we know that the domain of a function is the set of real values on which the given function is defined.
so, it is clear that the function is defined for all the real values greater than or equal to -4.
Hence, the domain of the given function { x | x ≥ -4 } & { y l y ≥ 2 } is
[ -4 , ∞ ).
Learn more about Functions here https://brainly.com/question/26209640
#SPJ1
Multiple-Choice Exam A student takes a 20-question, multiple-choice exam with two choices for each question and guesses on
each question. Assume the variable is binomial.
Find the probability of guessing at least 10 out of 20 correctly. Round the answer to at least four decimal places.
P(guessing at least 10 out of 20 correctly)
27
0
E
The probability is 0.5881.
From the question, we have
n = 20
P(X[tex]\geq[/tex]10) = 1-P(X<10)
P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)
P(X[tex]\leq[/tex]9) = P(X=0)+P(X=1)+..................+P(X=9)
=0.4119
P(X[tex]\geq[/tex]10) = 1-P(X[tex]\leq[/tex]9)
=1-0.4119
=0.5881
Probability:
Probability refers to possibility. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution. Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is.
To learn more about probability visit: https://brainly.com/question/11234923
#SPJ1
Tim has 53 boxcars. His grandfather gave him his entire collection from when he was a child. Use the following equation to find out how many cars Tim’s grandfather gave him: 1.6 x 102. How many cars does Tim have in all?
The number of cars that Tim has in all is 215.
How many cars does Tim have?The boxcars that Tim has is written in standard form. While the boxcars that Tim's grandfather gave him is written in scientific form. The first step is to convert the scientific form to standard form.
Number of boxcars that Tim's grandfather gave him: 1.62 x 10²
= 1.62 x 100
= 162
The next step is to add the boxcars that Tim's grandfather gave him to the number of boxcars that Tim has.
Total boxcars that Tim has = 162 + 53 = 215 box cars
To learn more about addition, please check: https://brainly.com/question/349488
#SPJ1
what is the answer for this question I am stuck
The area of triangle XYZ in the given graph with points X(-6,-5), Z(-4,-2) and Y(-2,-6) is 4 units square.
If the triangle's three vertices are specified on the coordinate plane, the area of a triangle in coordinate geometry may be determined. In coordinate geometry, a triangle's area is the area or space it occupies in the 2-D coordinate plane. The formula for finding the area of a triangle with three coordinates [tex](x_1,y_1), (x_2,y_2),[/tex] and [tex](x_3,y_3)[/tex] is as follows:
Area of triangle [tex]=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]
Now, finding the area of trianlge with points X(-6,-5), Z(-4,-2) and Y(-2,-6).
Area of triangle [tex]=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|[/tex]
[tex]=\frac{1}{2}|-6(-2+5)-4(-6+5)+-2(-5+2)|\\=\frac{1}{2}\times|-18+4+6|\\ =4[/tex]
Therefore, the area of triangle XYZ in the given graph is 4 units square.
To learn more about the Area of the triangle follow the below link:
https://brainly.com/question/19044159
#SPJ9
Ava went shopping for a new pair of pants the listed price of the pair of pants was $21 but the price with tax came out to $21.63 find the percent sales tax
The sales tax percent when Ava went shopping for a new pair of pants is 3%.
How to calculate the sales tax?From the information, Ava went shopping for a new pair of pants the listed price of the pair of pants was $21 but the price with tax came out to $21.63.
The sales tax will be:
= $21.63 - $21
= $0.63
The percentage of sales tax will be:
= Sales tax / Original price × 100
= 0.63 / 21 × 100
= 3%
The sales tax is 3%.
Learn more about percentages on:
brainly.com/question/24304697
#SPJ1
suppose that $36,000 is deposited in an account and the balance increases to $39,786.15 after 2.5 years. how long will it take for the account to grow to $51,806.67? Assume continous compounding
It will take about _ years for $36,000 to grow to $51,806.67
Answer:
About 9 years===================================
Continuous compounding equation:
[tex]P(t) = P_0e^{rt}[/tex]Given[tex]P_0=36000[/tex][tex]P(t)=39.876.15[/tex][tex]t=2.5[/tex]Find the value of r[tex]39786.15= 36000*e^{r*2.5}[/tex][tex]e^{2.5r}=39786.15/36000[/tex][tex]ln \ e^{2.5r}= ln \ 1.10517[/tex][tex]2.5r=0.099[/tex][tex]r=0.099/2.5[/tex][tex]r=0.04[/tex]Now find the time t for the balance to grow to $51806.67:
[tex]51806.67= 36000*e^{0.04t}[/tex][tex]51806.67/36000=e^{0.04t}[/tex][tex]e^{0.04t}= 1.439[/tex][tex]ln \ e^{0.04t}= ln \ 1.439[/tex][tex]0.04t=0.3639[/tex][tex]t=0.3639/0.04[/tex][tex]t=9.0975[/tex]The time required is about 9 years.
Answer:
9 years
Step-by-step explanation:
Continuous Compounding Formula
[tex]\large \text{$ \sf A=Pe^{rt} $}[/tex]
where:
A = Final amount.P = Principal amount.e = Euler's number (constant).r = Annual interest rate (in decimal form).t = Time (in years).Given values:
P = $36,000A = $39,786.15t = 2.5 yearsSubstitute the given values into the formula and solve for r to find the annual interest rate:
[tex]\implies \sf 39786.15=36000 \cdot e^{2.5r}[/tex]
[tex]\implies \sf \dfrac{39786.15}{36000}= e^{2.5r}[/tex]
[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=\ln e^{2.5r}[/tex]
[tex]\implies \sf \ln \left(\dfrac{39786.15}{36000}\right)=2.5r[/tex]
[tex]\implies \sf \dfrac{1}{2.5}\ln \left(\dfrac{39786.15}{36000}\right)=r[/tex]
[tex]\implies \sf r=\dfrac{2}{5}\ln \left(\dfrac{39786.15}{36000}\right)[/tex]
[tex]\implies \sf r=0.03999996933...[/tex]
[tex]\implies \sf r=0.04\;(2\;d.p.)[/tex]
Therefore, the annual interest rate is 0.04 = 4%.
To calculate how many years it will take for $36,000 to grow to $51,806.67, substitute the values into the formula and solve for t:
[tex]\implies \sf 51806.67=36000 \cdot e^{0.04t}[/tex]
[tex]\implies \sf \dfrac{51806.67}{36000}= e^{0.04t}[/tex]
[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= \ln e^{0.04t}[/tex]
[tex]\implies \sf \ln\left(\dfrac{51806.67}{36000}\right)= 0.04t[/tex]
[tex]\implies \sf \dfrac{\ln\left(\dfrac{51806.67}{36000}\right)}{0.04}=t[/tex]
[tex]\implies \sf t=9.10 \; (2\;d.p.)[/tex]
Therefore, it will take about 9 years for $36,000 to grow to $51,806,67.