Use two different methods to find an explain the formula for the area of a trapezoid that has parallel sides of length a and B and height H. Consider these ideas or others: combine two copies of a trapezoid to make a parallelogram. Subdivide the trapezoid into two triangles one with base b and one with base a.
Answers
Answer:
Formula of Trapezoid:
A = (a + b) × h / 2
The formula can be derived in different ways. for now, we have discussed two ways:
1. By using the formula of a triangle
2. By dividing into different sections
Step-by-step explanation:
1. By using the formula of a triangle
One of the ways to explain a formula for an area of a trapezoid using a formula for a triangle can be as follows.
Assume a trapezoid PQRS with lower base SR and upper base PQ (they are parallel) and sides PS and QR.
The image is attached below.
Connect vertices P and R with a diagonal.
Consider triangle ΔPQR as having a base PQ and an altitude from vertex R down to point M on base PQ (RM⊥PQ).
Its area is
S1=[tex]\frac{1}{2} *PQ*RM[/tex]
Consider triangle ΔPRS as having a base SR and an altitude from vertex P up to point N on-base SR (PN⊥SR).
Its area is
S2=[tex]\frac{1}{2} *SR*PN[/tex]
Altitudes RM and PN are equal and constitute the distance between two parallel bases PQ and SR.
They both are equal to the altitude of the trapezoid h.
Therefore, we can represent areas of our two triangles as
S1=[tex]\frac{1}{2}*PQ*h[/tex]
S2=[tex]\frac{1}{2}*SR*h[/tex]
Adding them together, we get the area of the whole trapezoid:
S=S1+S2=[tex]\frac{1}{2} (PQ+SR)h,[/tex]
which is usually represented in words as "half-sum of the bases times the altitude".
2. By dividing into different sections
Trapezoid PQRS is shown below, with PQ parallel to RS.
Figure 1 - Trapezoid PQRS with PQ parallel to RS(image is attached below.)
We are going to derive the area of a trapezoid by dividing it into different sections.
If we drop another line from Q, then we will have two altitudes namely PT and QU.
Figure 2 - Trapezoid PQRS divided into two triangles and a rectangle. (image is attached below.)
From Figure 2, it is clear that Area of PQRS = Area of PST + Area of PQUT + Area of QRU. We have learned that the area of a triangle is the product of its base and altitude divided by 2, and the area of a rectangle is the product of its length and width. Hence, we can easily compute the area of PQRS. It is clear that
=> [tex]A_{PQRS} = (\frac{ah}{2}) + b_{1}h + \frac{ch}{2}[/tex]
Simplifying, we have
=>[tex]A= \frac{ah+2b_{1+C} }{2}[/tex]
Factoring we have,
=> [tex]A_{PQRS} = (a+ 2b_{1} + c)\frac{h}{2} \\= > {(a+ b_{1} + c) + b_{1} }\frac{h}{2}[/tex]
But, [tex]a+ b_{1} + c[/tex] is equal to [tex]b_{2}[/tex], the longer base of our trapezoid.
Hence, [tex]A_{PQRS}= (b_{1} + b_{2} )\frac{h}{2}[/tex]
We have discussed two ways by which we can derive area of a trapezoid.
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Related Questions
A certain field is a rectangle with a perimeter of 1030 feet. The length is 181 feet more than the width. Find the width and length of the rectangular field.
Answers
Step-by-step explanation:
the perimeter is
2×length + 2×width = 1030 ft
length + width = 515 ft
length = width + 181
now we use that in the first equation :
(width + 181) + width = 515
2×width + 181 = 515
2×width = 334
width = 167 ft
because of
length = width + 181
length = 167 + 181 = 348 ft
The width and length of the rectangular field are 165.5 ft and 346.5 ft respectively.
Concept: Perimeter is the sum of the length of all sides of a figure/solid.
Given: Length = 181+width
Let width be equal to 'width' ft
Perimeter= 1030 feet
In a rectangle there are 2 length and 2 width so the perimeter of a rectangle is equal to 2*(181+ width) + 2width = 362 + 2width + 2 width = ⇒362 + 4width =1030 ft
4width = 662 ( on transposing)
width= 165.5 ft
length= 181+ 165.5 ft
length = 346.5 ft
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the model of an airplane has a wingspan of 20 inches. the model has a scale of 1 inch = 4 feet. what is the wingspan of the actual airplane? show your example
Answers
The wingspan of the actual aeroplane is 80 feet.
Given, that the model of an aeroplane has a wingspan of 20 inches and the scale factor is 1 inch = 4 feet.
We need to find what is the wingspan of the actual aeroplane.
What is a scale factor?The scale factor is a way to compare figures with commonly described but differing scales or measurements.
The wingspan of the actual aeroplane = The wingspan of the model aeroplane × Scale factor.
⇒The wingspan of the actual aeroplane = 20 × 4 feet = 80 feet.
Therefore, the wingspan of the actual aeroplane is 80 feet.
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Reuben rented a truck for one day. There was a base fee of $14.95, and there was an additional charge of 80 cents for each mile driven. Reuben had to pay
$135.75 when he returned the truck. For how many miles did he drive the truck?
Answers
Answer:
hczgdjatulsfulsitsfisfula7tl7roa7pts
What is x, the distance between points A and A'? D.15
Answers
The value of x that fills the coordinate are 7 or -5
Distance between two pointsThe formula for calculating the distance between two points is expressed as:
D = √(x2-x1)² +(y2-y1)²
Substitute the given coordinates
10 = √(15-7)² +(1-x)²
100 = 8² +(1-x)²
(1-x)² = 100 - 64
(1-x)² = 36
1- x = -6
x = 7 or -5
Hence the value of x that fills the coordinate are 7 or -5
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Complete question
If the distance between two points (x, 7) and (1, 15) is 10, find the value of x.
in 1979 the Wincom river was 33 feet below the bridge. Because of silt build-up in the river bottom, the
river was only 12 feet below the bridge by 1986. Write an equation for the distance of the river from the
bridge, d, with t = 0 representing 1979. If the silt build-up continues at the same rate, what year will the
river reach the bridge?
Answers
Answer: d = 40 - 5/2t; 1995
Hope it helps :)
A tower on a college campus was built with a faulty foundation and is starting to lean. A student climbs to the tilted top and drops a rope down to the ground. The
end of the rope lands 3 feet from the base of the tower and measures 57 feet from
the top of the building to the ground. What is the angle that the tower is leaning?
Round your answer to the nearest degree, and enter the number only
Answers
Answer:
leaning at 3 degrees off of normal (to the nearest whole degree)
Step-by-step explanation:
To visualize, let's assume the tower is leaning to the right, (as viewed by us).
If the rope is dropped from the top to the ground, the tower, the rope and the ground (between the rope and the tower) form a right triangle.
The height and base of the triangle are given
Often we're looking for the angle near the ground, but here, we're looking for the angle between the rope and the tower because the angle is congruent to the angle that the tower forms with it's original vertical position (alternate interior angles). If the tower was standing straight up and hadn't been leaning at all, we wouldn't say that the tower was leaning 90°... we would say it was leaning 0°. Whereas if the tower fell over and was laying flat on the ground, we would say it was leaning at 90°. So, we're not measuring the angle between the tower and the ground, but rather the angle between the tower where it should be, and where it is... which is the same angle as the rope forms with the tower.
Recall that [tex]tan(\theta)=\dfrac {opp}{hyp}[/tex]
Since we know the two legs of the right triangle, one can setup a tangent relationship with the two legs, but remember that the "opposite" side is going to be the ground, and the "adjacent side" will be the rope.
[tex]tan(\theta)=\dfrac {ground}{rope}\\tan(\theta)=\dfrac {3ft}{57ft}\\tan(\theta)=\frac {3}{57}\\\theta=tan^{-1}(\frac {3}{57})\\\theta=3.012787504^o\\\theta \approx 3^o\\[/tex]
If you bought a stock last year for a price of $103, and it has risen 13.5% since then, how much is the stock worth now, to the nearest cent?
Answers
The stock worth now, to the nearest cent is $90.75.
How to find the current worth of stock?The current worth of stock = initial price of stock+increase in the price of stocks
If you bought stock last year for a price of $103, and it has risen 13.5% since then,
Let X be the worth of stock;
103 × 100 = (13.5% + 100%) × X
10300 = 113.5X
X = 10300/113.5
X = 90.74889868
X = $90.75
Hence, The stock worth now, to the nearest cent is $90.75.
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what is 2 plus two.
please answer quickly
Answers
Answer:
2 + 2: 4
or 2 + 2: 5 from the dystopian novel 1984.
0, 1, 2, 3, 4, 5
Two plus two is a representative expression of the value initially being at 2, and adding two more values to the 2 that we already have. Since we are adding two to the quantity we already have, we can see that the answer is four (4) because we can simply count up two numbers from 2 to get what 2 + 2 equals. When counting up, we do not count the number that is already begun with because that would invalidate the equation. Since three (3) is after two (2), we can count starting at three. 3, 4! Those are the two values we get to when in the process of adding two. When we add one to two, we get three. When we add two to two, we get four. That is why 2 + 2 = 4. If you need help understanding better, let me know and I will gladly assist you!
Question 7 of 10 Which system of inequalities is graphed below? 10 -10 -8- 8 10 OA. y > X - 4 y < x +6 OB. VS X - 4 y < x +6 O c. y2 x -4 y > x + E OD. VS X - 4 y > x2 + 6
Answers
Because in both graphs the shaded region is above the curve, the correct option is C.
Which pair of inequalities is graphed?
On the graph we can see two parabolas, such that the y-intercepts of these are:
y = 6
y = -4
So we just need to find the pair of options with these y-intercepts, and noticing that for both cases, we must have:
y > parabola.
Because both shaded regions are above the parabolas.
The only option that meets these requirements is option C. Where there is a mistake, as you can see both curves have dashed lines, and always when we have dashed lines we should use the symbols > or <, while in that option we have a ≥ symbol
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pls help!!!
What is the range of the function on the graph?
all real numbers
all real numbers greater than or equal to 0
all real numbers greater than or equal to 1
all real numbers greater than or equal to 2
Answers
Answer:
All real numbers greater than or equal to 2.
Step-by-step explanation:
To find the range, we look at the y values. Since all the y values never dip under 2 and also keep increasing to infinity, the range is [tex]y\geq 2[/tex].
Answer:
all real numbers greater than or equal to 2
Step-by-step explanation:
Edge
It has been estimated that only about 40% of California residents have adequate earthquake supplies suppose you randomly survey 22 California residents we are interested in the number who have adequate earth quake supplies
Answers
The answers are as follows:
a) X denotes the number of the california residents that have adequate earthquake insurance
b) x = 1 ,2 ,3 ......
c) P( X=x ) = 0.3(1-0.3) ^ (x-1)
d) P( X=1) + P( X=2) + P( X=3) + P( X=4)
e) 0.49
f)0.42
g)0.33
What is probability?It is a branch of mathematics that deals with the occurrence of a random event.
The complete question is
It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose we are interested in the number of California residents we must survey until we find a resident who does not have adequate earthquake supplies. a. In words, define the random variable X. b. List the values that Xmay take on. c. Give the distribution of X.X~ _____(_____,_____) d. WhatistheprobabilitythatwemustsurveyjustoneortworesidentsuntilwefindaCaliforniaresidentwhodoes not have adequate earthquake supplies? e. What is the probability that we must survey at least three California residents until we find a California resident who does not have adequate earthquake supplies? f. HowmanyCaliforniaresidentsdoyouexpecttoneedtosurveyuntilyoufindaCaliforniaresidentwhodoesnot have adequate earthquake supplies? g. How many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies?
given that 30% of California residents have adequate earthquake supplies.
a) variable X denotes the number of the california residents that have adequate earthquake insurance
b) x = 1 ,2 ,3 ......
c) p=0.3
P( X=x ) = 0.3(1-0.3) ^ (x-1)
d) P( X=1) + P( X=2) + P( X=3) + P( X=4)
= 0.3(1-0.3) ^ 0 + 0.3(1-0.3) ^1 + 0.3(1-0.3)^ 2 +...
e) P( X≥ 3) = 1- P(X<3)
= 1- (P(X=1) + (X=2))
= 1- 0.051
= 0.49.
f) E(X)= 1/P
p is the resident who does not have adequate earthquake supplies
p= 1-0.3
p = 0.7
E(X) = 1/0.7
= 0.42
g) E(X) = 1/q
= 1/0.3
= 3.333
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i need to Match each equation that represents each situation.
Answers
Step-by-step explanation:
this is quite easy, when you use just common sense and identify the right numbers and variable names.
they is nothing special needed, you don't even need to create the equations yourself.
$4.99 per pound. buys b pounds and pays $14.95.
14.95 = 4.99 × b
$4.99 per pound. buys b pounds and pays c.
this is exactly the same as before, just that this time the total amount is not given as a constant but as a variable.
c = 4.99 × b
d dollars per pound. buys b pounds and pays t.
the same as the 2 cases before, just now everything is a variable. no more constants, but otherwise the completely same structure and method.
t = d × b
earned $275, which is $45 more than Noah ("n").
$275 = n + $45
earned m dollars, which is $45 more than Noah ("n").
m = n + $45
earned m dollars, which is y dollars more than Noah (we are asked that Noah's earnings are now called "v").
here your teacher made a mistake.
sure, the only remaining answer is
v = m + y
but it is not correct. given the names of the prime and associated variables the correct answer would be
m = v + y or v = m - y
Joshua has a job that pays $2,265 biweekly. How much is withheld annually for Medicare? A.$853.91 B.$32.84 C.$16.42 d.$35.58 E.$71.16
Answers
Answer:
09999kjht
Step-by-step explanation:
tttttttttgtgr312345678909ytrewq
Two forces, F1 and F2, are acting on a body, as shown in the figure. The resultant force, R, is
newtons.
Answers
Answer:
R ,is the Resultant of two forces,
∠ACB +50°=180°
∠ACB=180° -50°
∠ACB=130°
→Cos(130°)=Cos (180°-50°)= -cos 50°= -0.6427
Magnitude of resultant force =70.70 Newtons(Approx)
The sum of a number and 3 is less than 10 if x denotes the number what would be the possible values of x
x>13
x<13
x>7
x<7
Answers
Answer:
x<7
Step-by-step explanation:
Consider the number x (as mentioned in the question). Now carefully pay attention to the given condition
⇒"The sum of a number and 3 is less than 10"
Remember the notation of the term "less than" in mathematical context is indicated by "<" sign. Hence our inequality is
⇒ x+3<10
adding or subtracting any number from the both sides of an inequality DOESN'T change its direction. Thus, if we cancel out 3 from both sides it won't swap the inequality. Therefore
⇒ x+3-3<10-3
⇒ [tex]x<7[/tex]
In terms of interval notation
[tex]\implies x\in( -\infty,7)[/tex]
In conclusion
The answer is D) [tex]x < 7[/tex]
Let the number be x
The sum of number and 3
x+3Is less than 10
x+3<10Lets solve it
x<10-3x<7Hence
Option D is correct
The cost of renting a car is $47 /week plus $0.25 /mile traveled during that week. An equation to represent the cost would be y=47+0.25x , where x is the number of miles traveled.
Answers
Answer:
a. 61.25
b. 70
c. 212
Step-by-step explanation:
a. What is your cost if you travel 57 miles?
The cost is $ .
y=47+0.25x
y=47+0.25(57)
y=47+14.25
answer: 61.25
b. If your cost was $64.50, how many miles were you charged for traveling?
You were charged for traveling miles.
64.50-47=0.25x
17.50=1/4x
17.50*4
answer 70=x
c. Suppose you have a maximum of $100 to spend for the car rental. What would be the maximum number of miles you could travel?
The maximum number of miles you could travel is miles.
100-47
53=1/4x
answer : x= 212
find the area dimensions are in centimeters.
Answers
Answer:
[tex]9600cm^{2}[/tex]
Step-by-step explanation:
Get the area of the triangle and the rectangle, then subtract the triangle's area for the rectangle's
Which statement is true about the graphs of the two lines y=-8x-5/4and y=1/8x+4/5?
Answers
The statement first "The lines are perpendicular to each other because -8 and 1/8 are opposite reciprocal of each other" is correct.
What is a straight line?A straight line is a combination of endless points joined on both sides of the point.
The slope 'm' of any straight line is given by:
[tex]\rm m =\dfrac{y_2-y_1}{x_2-x_1}[/tex]
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
We have two lines given:
y = -8x - 5/4 ..(1)
y = x/8 + 4/5 ..(2)
Slope of the lines
m(1) = -8
m(2) = 1/8
m(1)m(2) = (-8)(1/8) = -1
Conditions for lines are perpendicular to each other.
Thus, the statement "The lines are perpendicular to each other because -8 and 1/8 are opposite reciprocal of each other" is correct.
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Which set of side lengths forms a right triangle?
O2, 3, 13
O4, 6, 10
O 9, 12, 18
O 15, 36, 39
Answers
Step-by-step explanation:
hope you can understand
Please help, been stuck on this for ages!
Answers
Answer:
[tex]\boxed {192 cm^{2}}[/tex]
Step-by-step explanation:
Find the volume of separate cuboids :
Volume (top cuboid) :
⇒ V = 2 x 6 x (10 - 2)
⇒ V = 12 x 8
⇒ V = 96 cm²
Volume (lower cuboid) :
⇒ V = 8 x 6 x 2
⇒ V = 48 x 2
⇒ V = 96 cm²
Volume (prism) :
⇒ 96 + 96
⇒ 192 cm²
Question 1
Kapoor Corporation uses the following activity rates from its activity-based costing to assign overhead costs to products.
Activity Cost Pools Activity Rate
Setting up batches $86.50 per batch
Assembling products $4.63 per assembly hour
Processing customer orders $52.16 per customer order
Data concerning two products appear below:
Product J00A Product S06U
Number of batches 34 43
Number of assembly hours 105 812
Number of customer orders 17 32
Required:
a. How much overhead cost would be assigned to Product J00A using the company's activity-based costing system?
b. How much overhead cost would be assigned to Product S06U using the company's activity-based costing system?
Overhead Cost:
There are some expenses incurred at the time of manufacturing of products and services that are not directly attributable to the finished products. These expenses are known as manufacturing overheads or indirect expenses.
Answers
The overhead cost would be assigned to Product J00A using the company's activity-based costing system is $4313.87.
How to calculate the cost?The overhead cost that would be assigned to Product J00A using the company's activity-based costing system will be:
= ($86.50 × 34) + ($4.63 × 105) + ($52.16 × 17)
= $2941 + $486.15 + $886.72
= $4313.87
The overhead cost would be assigned to Product S06U using the company's activity-based costing system will be:
= ($86.50 × 43) + ($4.63 × 812) + ($52.16 × 32)
= $3719.50 + $3759.56 + $1669.12
= $9148.18
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35. The grocery store sells a package of 6 pudding cups for $2.88. What is the cost for one of the pudding
cups in the package?
Answers
Answer:
48 cents
Step-by-step explanation:
Divide 2.88 by 6 to determine the value of a single pudding.
2.88/6 = 0.48
Hence, each pudding is worth 48 cents.
Hope this helps!
Calculate the gross earning for an apple picker based on the following differential pay scale: 1- 1000:$0.03 each 1001-1600:$0.06 each over 1600:$0.07 each. Calculate the Gross earning
Answers
If he picks 1965, then the Gross earning will be $85.55.
The missing part of the question is given below.
If he picks 1965, then Calculate the Gross earning.
What is Algebra?The analysis of mathematical representations is algebra, and the handling of those symbols is logic.
Calculate the gross earning for an apple picker based on the following differential pay scale:
1 – 1000: $0.03 each
1001 – 1600: $0.06 each
over 1600: $0.07 each
Total apples picked by Lindbergh = 1965
So, for 1000 apples, he will get =1000 × 0.03 = $30
Now, remaining apples = 1965 – 1000 = 965
Now for next 600 apples (from 1001-1600), he will get = 600 × 0.05 =$30
Now, remaining apples = 965 – 600 = 365
So, for remaining 365 (above 1600) apples, he will get = 365 × 0.07 = $25.55
So, total gross earning = $30 + $30 + $25.55
⇒ $85.55 is total gross earning
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I need help question 4
Answers
Answer:
-51
Step-by-step explanation:
If the limits are reversed, then place a negative sign in front of the integral.
[tex]\int\limits^a_bf ({x}) \, dx =-\int\limits^b_a f({x}) \, dx[/tex]
[tex]\int\limits^a_3 f({x}) \, dx = 51\\[/tex] where [tex]\alpha =10\\[/tex]
[tex]\int\limits^3_\alpha f( {x}) \, dx =-51[/tex] where[tex]\alpha =10[/tex]
Which of the following could be the first step in solving the equation below 4+2log_(3)x=17
Answers
Answer:
subract 4 from both sides
Step-by-step explanation:
Alina has $20 in her account she wants to purchase a purse shoes that cost $45 if the line makes the purchase which ener will represent the amount of money in the line of the account
Answers
The line represent the amount of money in the line of the account will be x + y = 45.
What is Algebra?The analysis of mathematical representations is algebra, and the handling of those symbols is logic.
Alina has $20 in her account she wants to purchase a purse, shoes that cost $45.
If the line makes the purchase, Then the line represent the amount of money in the line of the account will be
x + y = 45
where x be the price of purse and y be the price of shoe.
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Which of the following is a solution to this inequality?
y<2/3x+2
O (0, 3)
O (-3, 1)
O (3,5)
O (1, 2)
Answers
Answer: D
Step-by-step explanation:
We can substitute in each of the coordinate pairs.
A) We get 3 < (2/3)(0)+2, which is false.
B) We get 1 < (2/3)(-3)+2, which is false.
C) We get 5 < (2/3)(3)+2, which is false.
D) We get 2 < (2/3)(1)+2, which is true.
You invest $1600 in an account paying 5.4% interest compounded daily. What is the account's effective annual yield? Write as a decimal.
Answers
Answer:
For a principle amount of $1600 and rate 5.4%, the effective annual yield is 5.55% if interest is compounded annually.
Step-by-step explanation:
Concept: The annual percentage yield (APY) or the effective annual yield is the real rate of return earned on an investment, taking into account the effect of compounding interest.
The principle amount is $1600, rate is 5.4% and period is 365 days. The formula to find yield is [tex]Y = (1+\frac{r}{n} )^{n} -1[/tex]. All the values are to be inserted in the formula. Here n = 365 and r = 5.4%.
[tex]Y=(1+\frac{0.054}{365} )^{365} -1\\Y=(1.000148)^{365} -1\\Y=5.55%[/tex]
The annual yield is 5.55%.
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Make sure you click the image for full picture
Answers
Answer:
See below.
Step-by-step explanation:
19.
The spinner has 12 equal-sized sections.
1/3 of 12 = 4
A number that appears in exactly 4 sections has a probability of 1/3. The number 3 appears exactly 3 times.
A number that appears more than 4 times has a probability greater than 1/3. The number 0 appears exactly 5 times.
A number that appears fewer that 4 times has a probability of less than 1/3.
Both 1 and 2 appear fewer than 4 times.
Answer:
Probability greater than 1/3: 0
Probability equal to 1/3: 3
Probability less that 1/3: 1, 2
20.
The amount of concentrate is 12 gallons.
The total amount of drink is 40 gallons.
The fraction of the concentrate out of the total is 12/40.
12/40 = 0.3
0.3 of each gallon is concentrate.
Color the bottom up to 0.30
Jimmy Page puts $30 into a savings account that promises 1% interest
compounded quarterly. Which formula would you use to determine the
account balance in 25 years?
Answers
In 25 years the account balance will be $38.51 if Jimmy Page puts $30 into a savings account that promises 1% interest compounded quarterly.
What is compound interest?It is defined as the interest on the principal value or deposit and the interest which is gained on the principal value in the previous year.
We can calculate the compound interest using the below formula:
[tex]\rm A = P(1+\dfrac{r}{n})^{nt}[/tex]
Where A = Final amount
P = Principal amount
r = annual rate of interest
n = how many times interest is compounded per year
t = How long the money is deposited or borrowed (in years)
P = $30
r = 1% = 0.01
n = 4
[tex]\rm A = 30(1+\dfrac{0.01}{4})^{4\times25}[/tex]
After calculating:
A = $38.51
Thus, in 25 years the account balance will be $38.51 if Jimmy Page puts $30 into a savings account that promises 1% interest compounded quarterly.
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