The machine's book value at the end of year three is $164,000.
How to calculate the depreciation rate per machine hour?The total amount of hours used by the machine in the first three years is: 2,000 + 4,000 + 8,000 = 14,000 hours
To calculate the annual depreciation expense, we must first establish the depreciation rate per machine hour. The following is how the depreciation rate per machine hour is calculated:
Depreciation rate per machine hour = (machine cost - estimated residual value) / total number of machine hours estimated
Rate of depreciation per machine hour = ($500,000 - $20,000) / 20,000 machine hours
Depreciation per machine hour = $480,000 divided by 20,000 machine hours
Rate of depreciation per machine hour = $24 per machine hour
We may compute the depreciation expense for each year by using the depreciation rate per machine hour:
Depreciation expense in year one = depreciation rate per machine hour x hours used in the first year
Depreciation expense in year one = $24 x 2,000 hours
Depreciation expense for the first year = $48,000
Year 2 depreciation expense = Depreciation rate per machine hour multiplied by the number of hours consumed in year 2.
Depreciation expense for year 2 = $24 x 4,000 hours
Depreciation expense for year two = $96,000
Depreciation expense in year three = depreciation rate per machine hour x hours used in year three
Depreciation expense in year three = $24 x 8,000 hours
Depreciation expense during the third year = $192,000
The accumulated depreciation for the first three years is the sum of the following depreciation expenses:
Accumulated depreciation equals the sum of year one depreciation expense, year two depreciation expense, and year three depreciation expense.
Depreciation accumulated = $48,000 + $96,000 + $192,000
Depreciation accumulated = $336,000
To compute the machine's book value at the end of the year 3. We deduct the accumulated depreciation from the machine's cost:
Machine book value = machine cost minus accumulated depreciation
The machine's book value is $500,000 minus $336,000
The machine's book value is $164,000
As a result, the machine's book value at the end of year three is $164,000.
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Line t has equation y = -x - 2. Find the distance between l and the point D(-7, 0).
Round your answer to the nearest tenth.
the distance between line t and point D(-7, 0) is approximately 3.5 units.
what is distance ?
Distance is a measure of the amount of space between two objects or points. It is usually measured in units such as meters, kilometers, miles, or feet. Distance can be either a scalar quantity
In the given question,
To find the distance between line t and point D(-7, 0), we need to first find the point on line t that is closest to point D.
We can use the formula for the distance between a point and a line in the coordinate plane:
distance = |Ax + By + C| / √(A^2 + B^2)
where Ax + By + C is the equation of the line in standard form, and (x, y) is the coordinates of the point.
In this case, the equation of line t is y = -x - 2, which we can rewrite in standard form as x + y + 2 = 0.
Using the formula above, we have:
distance = |x + y + 2| / √(1^2 + (-1)^2)
distance = |(-7) + 0 + 2| / √(2)
distance = 5 / √(2)
distance ≈ 3.5 (rounded to the nearest tenth)
Therefore, the distance between line t and point D(-7, 0) is approximately 3.5 units.
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evaluating Outcomes with Probability: Tutorial Question Select the correct answer. Students voted in an election for class president, but there was a tie for first place between two people. The school principal suggested the following methods could be used to decide who the class president should be. Use probability to determine which method could be used to fairly choose the class president. O flipping a fair coin O asking a group of senior teachers O rolling a biased die Oletting the two candidates mutually decide who should win
Answer:
The method that could be used to fairly choose the class president in the given scenario is flipping a fair coin. This is because flipping a fair coin is a random process and both candidates have an equal chance of winning. This ensures fairness in the selection process. Asking a group of senior teachers or letting the two candidates mutually decide who should win may introduce biases and may not be fair to both candidates. Rolling a biased die may also introduce unfairness as the outcome is not random and is predetermined.
Find the measures of angle BED
keeping in mind that twin sides make twin angles, Check the picture below.
what’s answer
0.57
0.7
0.82
1.44
Answer:
Step-by-step explanation:
Sin O= opp/hyp
Sin 55 = BC/AB
Sin 55=.82
so
BC/AB=.82
In physical education class, students get one sticker for each mile they walk and two stickers for each mile they run. Jenny earned nine stickers
and completed seven miles. What would the graph look like?
The graph of the equation is linear
Given data ,
In physical education class, students get one sticker for each mile they walk and two stickers for each mile they run.
And , Jenny earned nine stickers
and completed seven miles
Based on the information provided, Jenny earned 9 stickers and completed 7 miles in physical education class. She receives one sticker for each mile she walks and two stickers for each mile she runs.
We can represent this information on a graph with the number of stickers earned on the vertical axis (y-axis) and the number of miles completed on the horizontal axis (x-axis).
Hence , the graph is solved
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A person invested $7600 for 1 year, part at 6%, part at 9%, and the remainder at 13%. The total annual income from these investments was $818. The amount of money invested at 13% was $1200 more than the amounts invested at 6% and 9% combined. Find the amount invested at each rate.
Step-by-step explanation:
Let X be the amount invested at 6%, Y be the amount invested at 9%, and Z be the amount invested at 13%.
From the problem, we know that:
X + Y + Z = 7600 ---(1) (the total amount invested is $7600)
0.06X + 0.09Y + 0.13Z = 818 ---(2) (the total income from the investments is $818)
Z = X + Y + 1200 ---(3) (the amount invested at 13% is $1200 more than the amounts invested at 6% and 9% combined)
We can use equation (3) to substitute for Z in equations (1) and (2), then solve for X and Y as follows:
X + Y + (X + Y + 1200) = 7600
2X + 2Y = 6400
X + Y = 3200
0.06X + 0.09Y + 0.13(X + Y + 1200) = 818
0.06X + 0.09Y + 0.13X + 0.13Y + 156 = 818
0.19X + 0.22Y = 662
Using the system of equations X + Y = 3200 and 0.19X + 0.22Y = 662, we can solve for X and Y to get:
X = 800
Y = 2400
Substituting back into equation (3), we get:
Z = X + Y + 1200 = 4400
Therefore, the amounts invested at 6%, 9%, and 13% were $800, $2400, and $4400 respectively.
****
Atte
THINK
What is the volume of the triangular prism shown below?
13 cm
8 cm
3 cm
The volume of the Triangular prism above is 156 cm^3
What is a Triangular Prism?
Triangular Prism is a three-dimensional shape consisting of two triangular ends connected by three rectangles. It has two identical triangles as parallel bases and if it is a right prism all lateral faces are rectangles.
How to determine this
The volume of a Triangular prism is calculated as
Volume = 1/2 * base * height * length
Where the Base = 8 cm
Height = 3 cm
Length = 13 cm
Volume = 1/2 * 8 * 3 *13
Volume = 1/2 *312
Volume = 156 cm^3
Therefore, the volume of the triangular prism is 156 cm^3
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Susan is going for a walk. She walks for 2 hours at a speed of 3.2 miles per hour. For how many miles does she walk?
Complete each ordered pair so that it is a solution of the given linear equation.
y=1/4×−8; (4, ), ( ,−11)
The first ordered pair is (4, )
The second ordered pair is ( ,-11)
Answer:
(4, -3)
(-28,-11)
Step-by-step explanation:
y = [tex]\frac{1}{4}[/tex] x - 4 Substitute 4 for x
y = [tex]\frac{1}{4}[/tex][tex](\frac{4}{1})[/tex] - 4 another name of 4 is [tex]\frac{4}{1}[/tex]
y = 1 - 4
y = -3
(4,-3)
y = [tex]\frac{1}{4}[/tex] x - 4 Substitute -11 for y
-11 = [tex]\frac{1}{4}[/tex] x - 4 Add 4 to both sides
-11 + 4 = [tex]\frac{1}{4}[/tex]x - 4 + 4
-7 = [tex]\frac{1}{4}[/tex]x Multiply both sides by 4
-7(4) = [tex]\frac{1}{4}[/tex][tex](\frac{4}{1})[/tex]x
-28 = x
(-28,-11)
Helping in the name of Jesus.
You are sent to the local tea shop to pick up 12 drinks. You purchase 8 sweet teas and 4 unsweetened teas. Unfortunately, you forgot to label them. If you pick 3 drinks at random, find the probability of each event below. Give your answers as simplified fractions.
a) All of the 3 drinks picked are sweet teas.
b) Exactly one drink is sweetened.
a) The probability of picking a sweet tea on the first draw is 8/12. Since we did not replace the first tea, the probability of picking another sweet tea on the second draw is 7/11. Similarly, the probability of picking a sweet tea on the third draw is 6/10. Therefore, the probability of picking 3 sweet teas in a row is:
(8/12) * (7/11) * (6/10) = 0.2545 or 127/500
b) There are 3 ways to pick exactly one sweet tea: S U U, U S U, U U S, where S represents a sweet tea and U represents an unsweetened tea. The probability of picking a sweet tea on the first draw is 8/12, and the probability of picking an unsweetened tea is 4/12. Therefore, the probability of picking exactly one sweet tea is:
(8/12) * (4/11) * (3/10) + (4/12) * (8/11) * (3/10) + (4/12) * (3/11) * (8/10) = 0.4364 or 48/110
Gerald is constructing a line parallel to line l through point P. He begins by drawing line m through points P and Q. He then draws a circle centered at Q, which intersects line l at point N and line m at point S. Keeping the compass measure, he draws a congruent circle centered at point P, which intersects line m at point T.
Which next step will create point R, such that when a line is drawn through points P and R, the line will be parallel to line l?
Lines m and n intersect at point Q. A circle is drawn around point Q and forms point S on line m and forms point N on line l. Point P is also on line m. A circle is drawn around point P and forms point T on line m.
To create point R such that a line through points P and R is parallel to line l, Gerald needs to draw a line through points T and N.
We have,
We know that line l is parallel to line m since both lines intersect at point Q and no other point.
The circle centered at Q intersects line m at point S and line l at point N, which means that QS is perpendicular to line l.
Similarly, the circle centered at P intersects line m at point T, which means PT is perpendicular to line l.
To create a line parallel to line l, Gerald needs to find a line perpendicular to line l.
This can be achieved by drawing a line through points T and N. This new line will be parallel to QS and perpendicular to line l.
Finally, Gerald can find the intersection of this new line with the circle centered at P to find point R, which will be equidistant from points P and T. Drawing a line through points P and R will be parallel to line l.
Thus,
To create point R such that a line through points P and R is parallel to line l, Gerald needs to draw a line through points T and N.
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if anyone understands this could u help me out???
Answer: V=1578.28 m³
Step-by-step explanation:
Volume is given as the formula
[tex]V=\frac{2}{3}\pi r^{3}[/tex]
r, radius, is the line from the center point to any end of the semi-sphere(that's what this shape is called) Here they show radius as
r=9.1
Substitute r into the formula
[tex]V=\frac{2}{3}\pi (9.1)^{3}[/tex] plug into calculator
V=1578.275 round to hundredths means 3 digits after the decimal point
V=1578.28 the number after the 7 is 5 or greater so you round up.
PLEASE HELP WILL GIVE BRAINLEST FOR CORRECT ANSWER ONLY !!
(Enlarge photo)
The answer is D :) )
Make a rap about why the kite is the best shape. Must be appropriate and at least 1 minute long. Giving brainless to whoever does it!!!!
Here is an attempt at an appropriately SFW 1 minute long rap about why the kite shape is the best:
The kite, soarin' high up in the sky,
Spreading colorful fabric far and wide.
There's no shape more perfect, this I must insist,
As it dances on the wind, ever blissful and lightly kissed.
Kite strings set it free, without any qualms,
Allowing it to float on high, away from harms.
Circling birds and beasts with graceful ease,
As a peaceful, wandering seas.
The kite shape is light but strong too,
Built to capture the breeze and soar true.
Diamond, square and box they can't compare,
To the kite so bright, beyond compare.
Many sizes to fit every quest,
From small pocket kites to giants they test.
Traditional or modern, simple or complex,
The kite is a form with many flexes.
Dragons and fighters, parafoils that stand,
On the beach or the park, across every land.
A kite for each mood and gusty seal,
As it dances and drifts from keel to keel.
So fly your kite up into the sun,
Some joyful moments, for now's the one.
Remember how free your kite did roam,
As a bird in the wind, and no longer alone.
The kite shape is art with physical grace,
Worth admiring for form, color and space.
There's magic up there among the open air,
As your kite soars high, beyond compare.
Here is an attempt at an appropriately SFW 1 minute long rap about why the kite shape is the best:
The kite, soarin' high up in the sky,
Spreading colorful fabric far and wide.
There's no shape more perfect, this I must insist,
As it dances on the wind, ever blissful and lightly kissed.
Kite strings set it free, without any qualms,
Allowing it to float on high, away from harms.
Circling birds and beasts with graceful ease,
As a peaceful, wandering seas.
The kite shape is light but strong too,
Built to capture the breeze and soar true.
Diamond, square and box they can't compare,
To the kite so bright, beyond compare.
Many sizes to fit every quest,
From small pocket kites to giants they test.
Traditional or modern, simple or complex,
The kite is a form with many flexes.
Dragons and fighters, parafoils that stand,
On the beach or the park, across every land.
A kite for each mood and gusty seal,
As it dances and drifts from keel to keel.
So fly your kite up into the sun,
Some joyful moments, for now's the one.
Remember how free your kite did roam,
As a bird in the wind, and no longer alone.
The kite shape is art with physical grace,
Worth admiring for form, color and space.
There's magic up there among the open air,
As your kite soars high, beyond compare.
Evaluate the expression shown below and write your answer as a fraction. -5/9 -(-9/4)
Step-by-step explanation:
When we simplify the expression -5/9 - (-9/4), we can rewrite it as:
-5/9 + 9/4
To add these fractions, we need to find a common denominator. The least common multiple of 9 and 4 is 36, so we can convert both fractions to have a denominator of 36:
-5/9 = -20/36
9/4 = 81/36
Now we can substitute these values into our expression and add them:
-20/36 + 81/36 = 61/36
Therefore, the simplified expression is 61/36.
A survey of the wine market has shown that the preferred wine for 17 percent of Americans is merlot.
A wine producer in California, where merlot is produced, believes the figure is higher in California. She
contacts a random sample of 550 California residents and asks which wine they purchase most often.
Suppose 115 replied that merlot was the primary wine.
(a) Calculate the appropriate test statistic to test the hypotheses.
(b) Calculate the p-value associated with the test statistic, and test the claim at α = 0.01.
The sample data provides sufficient evidence to conclude that the population proportion is greater than 0.17 at 1% level of significance.
What is a Null Hypothesis?
One can define a null hypothesis as a declaration that postulates no noteworthy variance or association between two or more variables existing within a populace.
Usually utilized in statistical hypothesis testing, it evaluates if an observed effect or relation is statistically meaningful or arises purely by chance.
It often bears the insignia H0 and subject to an alternative proposition (Ha) signifying a significant outcome or connection instead. Whenever the null hypothesis gets dismissed, one may deduce that there are ample findings to sustain the alternate assertion's validity.
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Supplementary angles
A point moves along a curve y=2x^2 + 1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x= 3/2?
If the "y-value" is decreasing at rate of 2 units per second, the rate at which "x" is changing when x=3/2 is -1/3 units per second.
The point moves along a curve having equation as : y = 2x² + 1; and
We know that y is decreasing at a rate of 2 units per second. We have to find the rate at which "x" is changing when x = 3/2,
To solve this problem, we differentiate, the curve equation,
So, taking the derivative of both sides of the equation with respect to time "t",
We get,
⇒ d/dt (y) = d/dt (2x² + 1),
⇒ dy/dt = (4x) × dx/dt,
We are given that dy/dt = -2 (since y is decreasing at a rate of 2 units per second), and we need to find "dx/dt" when x = 3/2,
Substituting "x = 3/2" and "dy/dt = -2",
We get,
⇒ -2 = 4×(3/2)(dx/dt),
⇒ -2 = (6)×(dx/dt),
⇒ dx/dt = -1/3,
Therefore, the rate at which "x" is changing when x = 3/2 is -1/3 units per second.
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The given question is incomplete, the complete question is
A point moves along a curve y=2x² + 1 in such a way that the y value is decreasing at the rate of 2 units per second. At what rate is x changing when x= 3/2 ?
More than 56% of people support stricter gun laws. Express the null and alternative hypotheses in symbolic form for this claim (enter as a percentage).
Null hypothesis: H0: p ≤ 0.56
Alternative hypothesis: H1: p > 0.56
We have,
Let p be the true proportion of people in the population who support stricter gun laws.
The null and alternative hypotheses in symbolic form for the claim that more than 56% of people support stricter gun laws are:
Null hypothesis: H0: p ≤ 0.56
Alternative hypothesis: H1: p > 0.56
Thus,
The null hypothesis states that the proportion of people in the population who support stricter gun laws is less than or equal to 56%, while the alternative hypothesis states that the proportion is greater than 56%.
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Solve completely the system of equations :
x + 3y - 22 = 0 , 2x - y + 42 = 0 , x - 11y + 142 = 0.
Answer:
x = -16/7 and y = 38/7.
Step-by-step explanation:
To solve the system of equations:
x + 3y - 22 = 0 --- equation (1)
2x - y + 42 = 0 --- equation (2)
x - 11y + 142 = 0 --- equation (3)
We will use the method of substitution to find the values of x and y that satisfy all three equations.
From equation (1), we can express x in terms of y:
x = 22 - 3y --- equation (4)
We can substitute equation (4) into equation (2) and simplify:
2(22 - 3y) - y + 42 = 0
44 - 6y - y + 42 = 0
-7y = -38
y = 38/7
Now, we can substitute the value of y into equation (4) to find x:
x = 22 - 3(38/7)
x = -16/7
Therefore, the solution to the system of equations is:
x = -16/7 and y = 38/7.
Write the following number in standard decimal form. one and ninety-six ten-thousandths 0 X
Use z scores to compare the given values.
<
The tallest living man at one time had a height of 222 cm. The shortest living man at that time had a height of 98.4 cm. Heights of men at that time had a mean of
171.59 cm and a standard deviation of 5.47 cm. Which of these two men had the height that was more extreme?
Since the z score for the tallest man is z- and the z score for the shortest man is z-. the
(Round to two decimal places.)
man had the height that was more extreme.
the more extreme height was the case for the shortest living man (13.38 standard deviation units below the population's mean) compare with the tallest man that was 9.21 standard deviation units above the population's mean.
To answer this question, we need to use standardized values, and we can obtain them using the formula:
z = (x - μ)σ .. [1]
Where,
x is the raw score we want to standardize.
μ is the population's mean.
σ is the population standard deviation.
A z-score "tells us" the distance from in standard deviation units, and a positive value indicates that the raw score is above the mean and a negative that the raw score is below the mean.
In a normal distribution, the more extreme values are those with bigger z-scores, above and below the mean. We also need to remember that the normal distribution is symmetrical.
Heights of men at that time had:
μ = 171.59 cm
σ = 5.47 cm.
Let us see the z-score for each case:
Case 1: The tallest living man at that time
The tallest man had a height of 222 cm.
Using [1], we have (without using units):
z = (x - μ)σ
z = (222 - 171.59)/5.47
z = 50.41 / 5.47
z = 9.21
That is, the tallest living man was 9.21 standard deviation units above the population's mean.
Case 2: The shortest living man at that time
The shortest man had a height of 98.4 cm.
Following the same procedure as before, we have:
z = (x - μ)σ
z = (98.4 - 171.59)/5.47
z = - 13.38
That is, the shortest living man was 13.38 standard deviation units below the population's mean (because of the negative value for the standardized value.)
The normal distribution is symmetrical (as we previously told). The height for the shortest man was at the other extreme of the normal distribution in standard deviation units more than the tallest man.
Then, the more extreme height was the case for the shortest living man (13.38 standard deviation units below the population's mean) compare with the tallest man that was 9.21 standard deviation units above the population's mean.
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The length if a photograph is 6 inches less than twice the width. The photograph is mounted in a frame that is 3 inches wide on all sides. If the area of the framed picture is 270 square inches, find the dimensions of the unframed photograph.
Answer:
see below
Step-by-step explanation:
L = 2W-6
given: (L+3)*(W+3) = 270
so (2W-6+6)*(W+6) = 270
so 2W*(W+6) = 270
open up the brackets
2w²+12W = 270
so solve quadratic equations of 2w²+12W - 270 =0
w = 9 or -15
so dimensions of the unframed photograph = width 9, length 12
Prove that (A-B)xC=(AXC)-(BXC)
Step-by-step explanation:
supposedly
A=3
B= -2
C=2
(3-(-2)×2=(3×2)-(-2×2)
5×2=6-(-4)
10=6+4
10=10
15 postcards, 10 envelopes, and a notepad cost 1 dollar and 68 cents. The envelope is 8 times cheaper than the notepad and 2 cents more expensive than the postcard. What is the price of the postcard, the envelope, and the notepad?
Let's start by setting up some equations to represent the given information.
Let's say the price of a postcard is "x" cents.
Then, according to the problem statement, the price of an envelope is 8 times cheaper than the notepad, which means the price of an envelope is (1/8)th of the price of the notepad. So the price of an envelope is (1/8)*y cents, where "y" is the price of the notepad in cents.
Also, we know that the price of an envelope is 2 cents more expensive than the price of a postcard, so we can write:
(1/8)*y = x + 2 ...(Equation 1)
We also know that there are 15 postcards and 10 envelopes in the purchase, so the total cost of the postcards and envelopes is:
15x + 10[(1/8)*y] ...(Equation 2)
Finally, we have a notepad that costs "y" cents. So the total cost of the purchase is:
15x + 10[(1/8)*y] + y ...(Equation 3)
We are given that the total cost of the purchase is 1 dollar and 68 cents, which is equal to 168 cents. So we can write:
15x + 10[(1/8)*y] + y = 168 ...(Equation 4)
Now we have four equations (Equation 1, Equation 2, Equation 3, and Equation 4) with three variables (x, y, and 168). We can solve for x and y by using a system of equations.
From Equation 1, we can solve for y in terms of x:
(1/8)*y = x + 2
y = 8x + 16
Substituting this expression for y into Equations 2 and 3, we get:
15x + 10[(1/8)*y] = 15x + 10(8x + 16) = 160x + 160
15x + 10[(1/8)*y] + y = 15x + 8x + 16 = 23x + 16
Substituting these expressions into Equation 4, we get:
23x + 16 = 168
Solving for x, we get:
x = 6
Substituting this value for x into Equation 1, we can solve for y:
(1/8)*y = x + 2 = 6 + 2 = 8
y = 64
So the price of a postcard is 6 cents, the price of an envelope is (1/8)*64 + 2 = 10 cents, and the price of a notepad is 64 cents
Wayne will toss a coin once and record eacthe toss as H for heads or T for tails.
Then he will randomly pick a card from a box that contains three cards numbered
1, 2, and 3, and he will record the number chosen. List all of the outcomes for the
event that the coin toss is tails.
Answer:
Step-by-step explanation:
The possible outcomes for the coin toss being tails and the card chosen being 1, 2, or 3 are:
-T1
-T2
-T3
Therefore, there are three possible outcomes for the event that the coin toss is tails.
Can someone please help me
The height of the Ferris wheel at point A is 100 ft.
The height of the Ferris wheel at point B is 182.1 ft.
The height of the Ferris wheel at point C is 39.9 ft.
What is the height of the Ferris wheel at each point?
The height of the Ferris wheel at each point is calculated as follows;
The height of a point on a circle; H = h + y
Where;
h is the height of the center of the circley is the vertical component of the point's position vectorH = h + rsinθ
where;
r is the radius of the circle.θ is the angle of rotationFor point A with θ = 0 radians;
H = 100 + 85 x sin (0)
H = 100 ft
For point B with θ = 7π/12 radians;
H = 100 + 85 x sin(7π/12)
H = 182.1 ft
For point C with θ = 5π/4 radians;
H = 100 + 85 x sin(5π/4)
H = 39.9 ft
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A laptop company has discovered their cost and revenue functions for each day:
C(x) = 4x²10x + 150 and R(x) = - 3x² +150x + 75. If they want to make a profit, what is the
range of laptops per day that they should produce? Round to the nearest nunber which would generate a
profit.
to
laptops
The range of laptops that they should produce per day to make profit is: 1 to 21 laptops
How to solve Inequality word problems?When dealing with inequalities, we could make use of any of the following:
Greater than(>)
Less than (<)
Greater than or equal to (≥)
Less than or equal to (≤)
We are given:
Cost function: C(x) = 4x² - 10x + 150
Revenue function: R(x) = -3x² + 150x + 75
Now, if they want to make profit, then:
R(x) > C(x)
Thus:
4x² - 10x + 150 > -3x² + 150x + 75
Rearranging to get:
7x² - 160x + 75 > 0
Solving using quadratic calculator gives:
x ≈ 1 and 21
Thus the range of laptops that they should produce per day to make profit 1 to 21
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Solve for the length of the missing side in the triangle. Leave your answer in radical form. Show your work and
explain the steps you used to solve.
17
The length of the missing side is given as follows:
[tex]x = \sqrt{155}[/tex]
What is the Pythagorean Theorem?The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
The theorem is expressed as follows:
c² = a² + b².
In which:
c is the length of the hypotenuse.a and b are the lengths of the other two sides (the legs) of the right-angled triangle.For the triangle in this problem, the sides and the hypotenuse are given as follows:
Sides of 13 and x.Hypotenuse of 18.Hence the missing side is given as follows:
x² + 13² = 18²
x = sqrt(18² - 13²)
x = sqrt(155) -> most simple radical form.
Missing InformationThe triangle is given by the image presented at the end of the answer.
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Suppose that $2000 is invested at an interest rate of 4.75% per year, compounded continuously. After how many years will the initial investment be doubled?
Answer: it will take approximately 14.62 years for the initial investment to double at an interest rate of 4.75% per year, compounded continuously.
Step-by-step explanation: Given an investment of $2000 at a continuously compounded interest rate of 4.75%, the balance in the account can be calculated using the following mathematical expression after t years:
The aforementioned equation, A = P * e^(rt), denotes the relationship between the accrued amount (A) and the principal amount (P), compounded continuously at a fixed annual rate of interest (r) over a specific time period (t), as governed by the mathematical constant "e."
In the context of financial calculations, the symbol 'P' denotes the initial capital investment. The interest rate, represented by the variable 'r', is expressed in the form of a decimal. Additionally, 'e' is the mathematical constant, roughly equivalent to 2.71828. Finally, 't' refers to the duration of the investment, measured in years.
In order to determine the duration of time required for the investment to achieve a twofold increase, it is necessary to solve the corresponding equation:
The equation expressed as 2P = P * e^(rt) can be restated more formally as follows. Given a principal investment amount represented by P and a rate of return indicated by r, compounded over time t, the equation can be expressed as the product of P and the exponential function of e^(rt), yielding twice the initial investment amount.
The variable 2P represents the monetary value acquired through doubling the initial investment.
Upon division of both sides by P, the resulting expression is as follows:
The equation 2 equals the exponential function of the base e raised to the power of the product of r and t.
By applying the natural logarithm function to both expressions, the resultant outcome is:
The natural logarithm of 2 can be represented as rt, where r denotes the logarithmic base and t denotes the logarithm of the argument, in accordance with the conventions of academic mathematical writing.
Upon resolving for the variable t, an outcome is yielded:
The mathematical expression t = ln(2) / r can be written in a formal academic style as follows: The equation determines the relationship between time t and the rate of decay r, where t is equal to the natural logarithm of 2 divided by r.
Upon substitution of the provided values, the resultant output is:
The calculated value of the variable t, representing the length of time in years, is approximately equal to 14.62 years, obtained through the algebraic manipulation of the natural logarithmic function of 2 divided by the constant value of 0.0475.