Answers
The expression (r-s)(x) is 4x² - x + 5, (r.s)(x) is 4x³ - 20x² and value of (r-s)(-1) is 10.
To find the expression (r-s)(x), we subtract the function s(x) from the function r(x):
(r-s)(x) = r(x) - s(x)
Substituting the given functions, we have:
(r-s)(x) = 4x² - (x-5)
Simplifying further:
(r-s)(x) = 4x² - x + 5
To find the expression (r.s)(x), we multiply the functions r(x) and s(x):
(r.s)(x) = r(x) × s(x)
Substituting the given functions, we have:
(r.s)(x) = (4x²)×(x-5)
Expanding and simplifying:
(r.s)(x) = 4x³ - 20x²
Now, let's evaluate (r-s)(-1) by substituting x = -1 into the expression (r-s)(x):
(r-s)(-1) = 4(-1)² - (-1) + 5
(r-s)(-1) = 4 - (-1) + 5
(r-s)(-1) = 4 + 1 + 5
(r-s)(-1) = 10
Therefore, (r-s)(-1) equals 10.
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Related Questions
2x + y ≤ 2
a. change the inequality into slope-intercept form, and then change it into an equation.
b. create a table. Use the x-values indicated
c. Graph the points and draw a line.
d. Shade the solution area.
e. Check your solution. Use (0,0) as your test point and see if it satisfies the conditions of the original inequality.
Answers
2x + y ≤ 2
y ≤ -2x + 2
To change it into an equation, we remove the inequality symbol and obtain:
y = -2x + 2
b. Using the x-values indicated, we can create a table of corresponding y-values:
|x | y |
|---|---|
|0 | 2 |
|1 | 0 |
|2 |-2 |
c. To graph the points and draw a line, we plot the points (0, 2), (1, 0), and (2, -2) on the coordinate plane and connect them with a straight line.
d. To shade the solution area, we need to determine which side of the line satisfies the inequality. Since the inequality is less than or equal to, we shade the area below the line.
e. To check the solution, we plug in the test point (0, 0) into the original inequality:
2x + y ≤ 2
2(0) + 0 ≤ 2
0 ≤ 2
The inequality holds, so the point (0, 0) is in the shaded area and satisfies the conditions of the original inequality. Therefore, the solution is valid.
Two math students were asked to write an exponential growth equation that had a starting value of 300 and a growth rate of 2%. Pierre thinks the answer is y=300(1.02)^x and Scott thinks that the answer is y=300(1.2)^x. Are either of them right and why?
Answers
Incorrect, the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation
Neither Pierre nor Scott has the correct exponential growth equation.
The exponential growth equation represents a relationship where a quantity increases or grows exponentially over time. It is typically represented as y = a(1 + r)^x, where "a" represents the initial or starting value, "r" represents the growth rate (expressed as a decimal), "x" represents the time or number of periods, and "y" represents the resulting value after the growth.
In this case, Pierre's equation is y =[tex]300(1.02)^x.[/tex]This equation suggests a growth rate of 2% (0.02 as a decimal), which means that the quantity would increase by 2% with each period. This aligns with the given growth rate of 2%. Thus, Pierre's equation is correct.
On the other hand, Scott's equation is y = [tex]300(1.2)^x[/tex]. This equation suggests a growth rate of 20% (0.2 as a decimal), which means that the quantity would increase by 20% with each period. However, the given growth rate is 2%, not 20%. Therefore, Scott's equation is incorrect.
To summarize, Pierre's equation, y = 300(1.02)^x, is the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation, y = 300(1.2)^x, does not match the given growth rate and is therefore incorrect.
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Find the area of each shape
Answers
The area of a triangle is 24 square feet and the area of a trapezium is 50 square feet.
From the given figure, the dimensions of trapezium are parallel sides a=7 ft, b= 13 ft and height = 5 ft.
Here, area of trapezium = 1/2 (Sum of parallel sides)×Height
= 1/2 ×(7+13)×5
= 1/2 ×20×5
= 50 square feet
Dimensions of triangle are base = 8 ft and height = 6 ft
Area of a triangle = 1/2 ×8×6
= 24 square feet
Therefore, the area of a triangle is 24 square feet and the area of a trapezium is 50 square feet.
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What is the equation for this line?
Answers
y=2/3x-4 is the equation of the line where 2/3 is the slope.
We have to find the equation of line which is passing through points (0, -4) and (3, -2).
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁.
Slope = -2+4/3-0
=2/3
Now let us find the y intercept.
-4=2/3(0)+b
b=-4
The equation of line is y=2/3x-4.
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find the invoice total, the amount should be paid after cash discount, the total amount due including shipping and insurance.
Answers
The total invoice is $788.80, we find this by find the cost of each item.
To find the invoice total, we need to calculate the total cost of each item.
For the tablecloth:
Quantity: 16
Unit Price: $35.00
Total Cost = Quantity × Unit Price = 16 × $35.00 = $560.00
For the rings:
Quantity: 8
Unit Price: $6.50
Total Cost = Quantity × Unit Price = 8 × $6.50 = $52.00
For the cups (cases):
Quantity: 4 cases
Unit Price: $25.30
Total Cost = Quantity × Unit Price = 4 × $25.30 = $101.20
For the bowls:
Quantity: 12
Unit Price: $6.30
Total Cost = Quantity × Unit Price = 12 × $6.30 = $75.60
Now, let's calculate the invoice total by adding up the total costs of each item:
Invoice Total = $560.00 + $52.00 + $101.20 + $75.60
= $788.80
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TriangleTriangle WXY with vertices W(8,-8), X(5,-2), and Y(3,-4) is drawn inside a rectangle, What is the area, in square units, of triangle LMN?
Answers
The area of the triangle whose dimensions are given in a vertices form would be = 9
How to calculate the area of a triangle given above?To calculate the area of the triangle, the formula that should be used is the coordinate geometry formula: area = the absolute value of Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By) divided by 2
Where; W=A, X= B, Y = C
Ax= 8
By= -2
Cy= -4
BX= 5
Ay = -8
CX= 3
That is:
= 8(-2--4)+5(-4--8)+3(-8--2)/2
= 16+20-18/2
= 18/2
= 9
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3500pounds is placed into a savings account that pays interest at arate of 1.5% compounded annually.
Calculate the total compound amount at the end six years of investment, rounded to 2 decimal places.
Answers
Answer:
$3827.00
Step-by-step explanation:
To calculate the total compound amount at the end of six years with an interest rate of 1.5% compounded annually, we can use the formula for compound interest:
A = P * (1 + r/n)^(n*t)
Where:
A = Total compound amount
P = Principal amount (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case:
P = $3500
r = 1.5% = 0.015 (as a decimal)
n = 1 (compounded annually)
t = 6 years
Substituting these values into the formula, we have:
A = 3500 * (1 + 0.015/1)^(1*6)
Calculating this expression, we get:
A ≈ 3500 * (1.015)^6 ≈ 3500 * 1.093717 ≈ 3827.00
Rounding the total compound amount to 2 decimal places, the final value is approximately $3827.00
Graph shows two triangles plotted on a coordinate plane. One triangle is at A (minus 4, 4), B (minus 8, 2), and C (minus 6, minus 6). Another triangle is at A prime (4, minus 2), B prime (2, 2), and C prime (6, 0). A sequence of transformations maps ∆ABC to ∆A′B′C′. The sequence of transformations that maps ∆ABC onto ∆A′B′C′ is a followed by a .
Answers
The sequence of transformations that maps triangle ABC onto triangle A'B'C' is a translation by (+8, -6) followed by a reflection across the y-axis.
To determine the sequence of transformations that maps triangle ABC onto triangle A'B'C', let's analyze the given coordinate points.
Triangle ABC:
A (-4, 4)
B (-8, 2)
C (-6, -6)
Triangle A'B'C':
A' (4, -2)
B' (2, 2)
C' (6, 0)
By comparing the corresponding vertices, we can identify the transformations:
Translation:
To transform A to A', we can observe that A' is obtained by moving 8 units to the right and 6 units downwards from A. Therefore, the first transformation is a translation by (+8, -6).
Reflection:
Next, we notice that B' is the reflection of B across the y-axis. Thus, the second transformation is a reflection about the y-axis.
Therefore, the sequence of transformations that maps triangle ABC onto triangle A'B'C' is a translation by (+8, -6) followed by a reflection across the y-axis.
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50 points, pls answer
Is it possible for a satellite to see
of the Earth’s equator? Why or why not?
11-94.
Answers
a. The image is attached below.
b. Length of the equator is 24,000 miles.
c. The image is attached below.
d. No, it is not possible for a satellite to see 50% of the Earth's equator.
Satellite B will see more of the Earth's equator than Satellite A. This is because the viewing angle of Satellite B is larger than the viewing angle of Satellite A. The viewing angle of Satellite B is 60 degrees, while the viewing angle of Satellite A is 45 degrees.
The length of the equator in view of Satellite B is 24,000 miles. This can be calculated using the following formula:
length of the equator = 2 * [tex]\Pi[/tex] * radius * sin(m/2)
Use a third color to draw the viewing angle from Satellite C. Label the points of tangency J and K. If m/C=45", find the mJK and mJZK
This is because the Earth is a sphere, and a sphere has no flat surfaces. A satellite can only see a portion of the Earth's surface, and the portion that it sees will always be less than 50% of the total surface area.
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9x⁸y² - 6x³y⁴ /3xy simplified is:
a. 3x⁹y³- 2x⁴y⁵
b. 3x⁷y - 2x²y³
c. 3x⁹y³ - 2x⁴y⁵
d. None of these choices are correct.
Answers
Answer: B
Step-by-step explanation:
The coefficients 9 & 6, divided by 3, equal 3 & 2.
For the exponents, if you divide the entire equation by xy, you need to subtract the corresponding exponents of x and y. Subtract the denominator's exponent from the numerator's exponent.
So, for the first part x^8/x^1=x^7
y^2/y^1=y
For the second part: x^3/x=x^2
y^4/y=y^3
So, if you put the coefficients and both parts of the equation together, you get 3x^7y - 2x^2y^3 (answer letter B).
Hope this helped!
The three steps below were used to find the value of the expression [(-10 + 2) - 1] + (2 + 3). Step 1: ? Step 2: -9 + 2 + 3 Step 3: -7 + 3 Which expression is missing from Step 1? Question 3 options: [-10 + -1 + 2] + (2 + 3) [-8 - 1] + (2 + 3) [-10 + 1] + (2 + 3) [8 + 1] + (2 + 3)
Answers
Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).
In order to find the missing expression in Step 1, let's analyze the given steps and the final expression.
Step 1: ?
Step 2: -9 + 2 + 3
Step 3: -7 + 3
To find the missing expression in Step 1, we need to work backwards from Step 3 to Step 1.
In Step 3, the expression "-7 + 3" gives us a result of -4.
In Step 2, the expression "-9 + 2 + 3" gives us a result of -4.
So, the missing expression in Step 1 should also evaluate to -4 when performed correctly.
Let's check the available options:
[-10 + -1 + 2] + (2 + 3) = -11 + 2 + 5 = -4
[-8 - 1] + (2 + 3) = -9 + 5 = -4
[-10 + 1] + (2 + 3) = -9 + 5 = -4
[8 + 1] + (2 + 3) = 9 + 5 = 14
Out of the given options, only option 2, [-8 - 1] + (2 + 3), correctly evaluates to -4. Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).
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In the right △ABC with m∠C=90°, m∠B=75°, and AB=12 cm. Find the area of △ABC.
Don't use trig to solve, don't know how to.
Answers
The area of the right triangle is 18.02 cm².
How to find area of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The sum of angles in a triangle is 180 degrees.
Therefore, let's find the area of the right angle triangle as follows;
let's find the height and base of the triangle.
using trigonometric ratios,
cos 75 = adjacent / hypotenuse
cos 75 = base / 12
base = 3.10582854123
Therefore,
sin 75 = opposite / hypotenuse
sin 75 = h / 12
height = 11.5911099155
height = 11.59
Therefore,
area of the triangle = 1 / 2 × 3.11 × 11.59
area of the triangle = 36.0483518371 / 2
area of the triangle = 18.0241759186
area of the triangle = 18.02 cm²
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Calculate the weight of a bed if its mass is 120 kg and gravitational acceleration is 20m/s2. Use weight equation.
Answers
Answer:
2400 N (Newtons)
Step-by-step explanation:
The weight of an object can be calculated using the equation:
Weight = mass * gravitational acceleration
Given:
Mass of the bed (m) = 120 kg
Gravitational acceleration (g) = 20 m/s²
Using the weight equation:
Weight = mass * gravitational acceleration
Weight = 120 kg * 20 m/s²
Weight = 2400 kg·m/s²
The unit of the weight is kilogram-meter per second squared (kg·m/s²), which is equivalent to the unit of force called Newton (N).
Therefore, the weight of the bed is 2400 Newtons (N).
100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
Answers
Answer:
A)19.0
B)10.0
Step-by-step explanation:
Using [tex]\frac{1}{2} * absin(C)[/tex]
This is used when you have an angle between two sides
Kindly check the attached image for the solution
Create TWO equivalent expressions for the following.
14(8−16x)+3x
Answers
Two equivalent expressions for the given expression 14(8 - 16x) + 3x are 112 - 221x and 112 - 221x.
Equivalent expression 1:
Expanding the expression 14(8 - 16x) and combining like terms, we get:
112 - 224x + 3x
Simplifying further, we have:
112 - 221x
Equivalent expression 2:
Distributing the coefficient 14 to both terms inside the parentheses, we have:
112 - 224x + 3x
Combining the terms with the same variable, we get:
112 - 221x
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Can somebody please help me thank tou
Answers
Answer:
Step-by-step explanation: On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer.
Answer:
Step-by-step explanation:
I Think this is the answer
On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer
Pleaseseee help
Two one-step equations
Two equations that contains fractions
One equation with distributive property
One equation with decimals
One real-world problem that is solved by an equation
Remember that each equation must include at least one variable
Answers
The correct equations are:
[tex]3x + 2 = 11 \\\[5y - 7 = 18\][/tex][tex]\frac{2}{3}x - \frac{1}{4} = \frac{5}{6}\\\frac{3}{5}y + \frac{2}{7} = \frac{1}{3}[/tex][tex]\[2(4x - 3) = 10\][/tex][tex]\[0.5x + 0.25 = 1.75\][/tex]1. Two one-step equations:
[tex]\[3x + 2 = 11\]\[5y - 7 = 18\][/tex]
2. Two equations that contain fractions:
[tex]\[\frac{2}{3}x - \frac{1}{4} = \frac{5}{6}\]\[\frac{3}{5}y + \frac{2}{7} = \frac{1}{3}\][/tex]
3. One equation with distributive property:
[tex]\[2(4x - 3) = 10\][/tex]
4. One equation with decimals:
[tex]\[0.5x + 0.25 = 1.75\][/tex]
5. Real-world problem solved by an equation:
A bakery sells cakes for $[tex]15[/tex] each. Let's say the total cost of cakes sold in a day is $[tex]180[/tex]. We can use the equation [tex]\(15x = 180\)[/tex] to find the number of cakes sold, represented by the variable [tex]x[/tex]. Solving the equation, we find [tex](x = 12\)[/tex]), indicating that the bakery sold [tex]12[/tex] cakes that day.
Here's a basic explanation for the real-world problem:
Imagine there is a bakery that sells cakes for $[tex]15[/tex]each. We want to find out how many cakes the bakery sold in a day if the total revenue from cake sales is $[tex]180[/tex]. To solve this problem, we can use an equation. Let's represent the number of cakes sold as [tex]x[/tex].
The equation [tex]\(15x = 180\)[/tex] is used to express that the total cost of the cakes sold [tex](\$15\ per \ cake)[/tex] is equal to $[tex]180[/tex]. To solve the equation, we divide both sides by [tex]15[/tex] to isolate the variable [tex]x[/tex]. The equation simplifies to [tex]\(x = 12\),[/tex] which means that the bakery sold [tex]12[/tex] cakes that day.
By using the equation, we can determine the number of cakes sold based on the given information and calculate the desired result.
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PLEASE ANSWER ASAP!!!!!
Answers
Answer:
[tex]\huge\boxed{\sf r = 5}[/tex]
Step-by-step explanation:
Given that,
7(q + 5) = (q + r)7Distribute7q + 35 = 7q + 7r
Subtract 7q from both sides7q - 7q + 35 = 7q - 7q + 7r
35 = 7r
Divide both sides by 735/7 = r
5 = r
OR
r = 5[tex]\rule[225]{225}{2}[/tex]
Answer:
r = 5
Step-by-step explanation:
Given statement,
→ 7(q + 5) is equivalent to (q + r)7.
Forming the equation,
→ 7(q + 5) = 7(q + r)
Now we have to,
→ Find the required value of r.
Then the value of r will be,
→ 7(q + 5) = 7(q + r)
Applying Distributive property:
→ 7(q) + 7(5) = 7(q) + 7(r)
→ 7q + 35 = 7q + 7r
Cancelling 7q from both sides:
→ 35 = 7r
→ 7r = 35
Dividing the RHS with number 7:
→ r = 35/7
→ [ r = 5 ]
Therefore, the value of r is 5.
What is the meaning of "[tex]dom(R)\subset \cup\cup R[/tex]"?
Answers
"dom(R) ⊂ UU R" means that the domain of relation R is a subset of the universal set associated with relation R
Understanding Set NotationThe notation "dom(R) ⊂ UU R" refers to the domain of a relation R being a subset of the universal set U.
Let me help you to break it down
1. dom(R): This represents the domain of the relation R. The domain of a relation is the set of all elements that appear as the first component in any ordered pair of the relation.
2. ⊂: This symbol indicates a subset relationship, meaning that the set on the left-hand side is a subset of the set on the right-hand side. In this case, "dom(R) ⊂ U" implies that the domain of relation R is a subset of the universal set U.
3. "UU R": The term "UU R" likely represents the universal set associated with relation R. It indicates the set that contains all possible elements that can be related by R.
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find the surface area
Answers
The surface area of the cylinder is: 156π.
Here, we have,
given that,
the cylinder has:
radius = 6 in
height = 10 in
now, surface area of the cylinder is:
SA = 2πrh + πr²
here, we have,
SA = 2π *6 * 10 + π6²
= 120π + 36π
= 156π
Hence, The surface area of the cylinder is: 156π.
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find the surface area
Answers
The Surface area of Triangular Prism is 132 cm².
We have have the dimension of prism as
Sides = 3 cm, 4 cm, 5 cm
and, l = 10 cm
and, b= 4 cm
Now, Surface area of Triangular Prism as
= (sum of sides) l + bh
= (3 + 4 + 5)10 + 4 x 3
= 12 x 10 + 12
= 120 + 12
= 132 cm²
Thus, the Surface area of Triangular Prism is 132 cm².
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7x 2/9 what the answer
Answers
Answer:
1.55
Step-by-step explanation:
2/9=0.22 x 7 =1.55
i hope this helps!
Taylor recorded Weekly grocery expense for the past 16 weeks and determine the mean weekly expense was $83.20 later she discovers that one week expense of $90 was incorrectly recorded at $38. What is the mean
Answers
The mean weekly expense, after correcting the incorrectly recorded week, is $81.25.
How to solveGiven that Taylor recorded the past 16 weeks' expenses, excluding the incorrect recording of $90 as $38, we have 15 correct expense values.
Sum of correct expenses = $83.20 * 15 = $1248
Now we need to include the corrected expense of $90 instead of $38.
New sum of expenses = $1248 - $38 (incorrectly recorded) + $90 (corrected) = $1300
Finally, we calculate the mean by dividing the new sum by the total number of weeks, which is 16.
Mean weekly expense = $1300 / 16 = $81.25
Therefore, the mean weekly expense, after correcting the incorrect recording, is $81.25.
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A dance team is made up of 9 girls and 3 boys. Two of the girls and 1 of the boys are also on the debate team. What percent of the dance team is also on the debate team? Enter your answer in the box.
Answers
Answer:
25%
Step-by-step explanation:
9+3=12 dance people total
divide 3 by 12 to determine debate members over dance members
=.25 or 25%
1. What will the coordinates of Triangle RST be after it is translated 4 units down and 3 units left?
--5-5
R
Da
4
64
5-
4
3-
bo
24
4
10
=arpy 5
-3
--5-
1 2 3 4 5 6 x
O Re(-7,-2), Sc(2, -1), Tc(-1,-9)
ORE(-1,-2), Sc(8,-1), Tc (5,-9)
ORE(-1,6), Sc(8,7), TC(5, -1)
O Re(-7,-2), Sc(8,-1), Tc(-1,-1)
Answers
The coordinates of the Triangle RST after it is translated 4 units down and 3 units to the left is the option;
R¢(-7, -2), S¢(2, -1), T¢(-1, -9)
What is a translation transformation?A translation transformation is one in which the location of the pre-image is transformed from its initial location to the destination location.
The coordinates of the vertices of the triangle RST are; R(-4, 2), S(5, 3), T(2, -5)
The translation transformation that is applied on the triangle RST includes;
A vertical translation 4 units downwards
An horizontal translation 3 units to the left
The translation transformation is therefore; T₍₋₃, ₋₄₎
Therefore, applying each transformation, we get;
R(-4, 2) ⇒ T₍₋₃, ₋₄₎ ⇒ R¢(-4 -3, 2 - 4) = R¢(-7, -2)
S(5, 3) ⇒ T₍₋₃, ₋₄₎ ⇒S¢(5 - 3, 3 - 4) = S¢(2, -1)
T(2, -5) ⇒ T₍₋₃, ₋₄₎ ⇒ T¢(2 - 3, -5 - 4) = T¢(-1, -9)
The correct option is therefore, the first option
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On an exam students are asked to list 5 historical events in the order in which they occurred. A student randomly orders the events. what is the probability that the
Answers
The probability that the student chooses the correct order is 1/120 or approximately 0.0083, which is equivalent to 0.83%.
How to solveIf the student randomly orders the events, there is only one correct order out of all possible permutations.
Hence, the likelihood of selecting the accurate sequence is equal to 1 over the total count of potential arrangements.
As there are 5 occurrences, the overall count of conceivable sequences is equivalent to 5 factorial (5!), amounting to 120.
Therefore, the probability that the student chooses the correct order is 1/120 or approximately 0.0083, which is equivalent to 0.83%.
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On an exam, students are asked to list 5 historical events in the order in which they occurred. A student randomly orders the events. What is the probability that the student chooses the correct order?
100 Points! Geometry question. Photo attached. Name the angle of depression and the angle of elevation in each figure. Thank you!
Answers
Answer:
carnival one
elevation = SWT
depression = RTW
deer one
elevation = ABC
depression = DCB
Step-by-step explanation:
If (2x-7), (x-1), 3x, x, (x+2), 30 and 10
1. Find the value of x
2. Find the value of the largest angle
PLEASE HELP ME ASAP
Answers
The value of x in arithmetic progression is 5/3 and the largest angle is 30.
The given sequence is: (2x-7), (x-1), 3x, x, (x+2), 30, 10.
In an arithmetic progression, the common difference (d) between consecutive terms remains constant.
The common difference between the terms = (x - 1) - (2x - 7) = x - 1 - 2x + 7 = 6 - x
6 - x = 3x - (x - 1)
6 - x = 3x - x + 1
6 - x = 2x + 1
-x - 2x = 1 - 6
-3x = -5
x = -5 / -3
x = 5/3
Therefore, the value of x is 5/3.
2x-7 = 2 ×5/3 -7 = -3.66
x-1 = 5/3-7 = -5.33
3x = 3 × 5/3 = 5
x + 2 = 5/3+2 =3.66
Thus, the largest angle is 30.
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Your question is incomplete, most probably the full answer is:
If (2x-7), (x-1), 3x, x, (x+2), 30 and 10 are in AP, then
1. Find the value of x
2. Find the value of the largest angle
There are books on a shelf. of these books are new. The rest of them are used. what is the ratio of used books to all books on the shelf?
Answers
Let's denote the total number of books on the shelf as 'total_books', and the number of new books as 'new_books'.
The number of used books can be calculated as the difference between the total number of books and the number of new books:
used_books = total_books - new_books
To find the ratio of used books to all books on the shelf, we divide the number of used books by the total number of books:
Ratio of used books to all books = used_books / total_books
Substituting the expression for used_books, we have:
Ratio of used books to all books = (total_books - new_books) / total_books
Simplifying further:
Ratio of used books to all books = 1 - (new_books / total_books)
Therefore, the ratio of used books to all books on the shelf is equal to 1 minus the ratio of new books to total books.
The perimeter of an isosceles triangle is 45 inches. Two sides of the triangle are congruent,
and the third side is twice as long as the sum of the congruent sides. What is the length of the
third side of the triangle in inches?
Answers
Answer:
Length of third side = 30 inches
Step-by-step explanation:
We will need a system of equations to find the length of the third side.
First equation:
Let x represent the length of just one of the congruent sides and let y represent the length of the third side.We know the perimeter of a triangle is simply the sum of its sides.
Thus, since the perimeter is 45, our first equation is x + x + y = 45, which simplifies to 2x + y = 45.
Second equation:
Since we're told that the length of the third side is twice as long as the sum of the congruent sides, our second equation is y = 2(x + x), which simplifies to y = 2(2x) and then finally y = 4x
Method: Substitution.
We can first solve for x (length of one of the congruent sides) by substituting y = 4x for y in 2x + y = 45:
2x + 4x = 45
6x = 45
x = 7.5
Find y:
Now we can find y (length of the third side) by plugging in 7.5 for x in y = 4x:
y = 4(7.5)
y = 30
Thus, the length of the third side is 30 inches.
Optional Step to check validity of answers.
First equation: Check that the sum of 7.5, 7.5, and 30 = 45
7.5 + 7.5 + 30 = 45
15 + 30 = 45
45 = 45
Second equation: Check that 30 is twice the sum of 7.5 + 7.5
30 = 2(7.5 + 7.5)
30 = 2(15)
30 = 30
Thus, our answers are correct and we've correctly determined the length of the third side of the triangle in inches.