Paul's rent budget is $750, given that his food budget of $225, is 30 percent of his rent budget.
What are fractions?Fractions are representation of quantities as a part of a certain quantity.
Written as a/b, read as a by b, and functions as a divided by b, represents a fraction showing a parts of b equal parts.
a is called the numerator, and b is called the denominator.
What are percentages?Percentages are modified fractions where the whole quantity is represented by 100.
To convert a fraction into percentages we multiply it by 100, while to convert percentages to fraction divided by 100.
How to solve the question?In the question, we are given that Paul's budget for food is 30% of the budget he has for rent.
We are asked if his budget for food is $225, what is his budget for rent.
We assume Paul's budget for rent to be $x.
We know that Paul's food's budget is 30 percent of his rent.
Hence, Paul's food budget = 30% of x.
or, Paul's budget for food = 30/100 * x {To convert percentages to fraction divided by 100}
But, we know that Paul's food budget is $225. Thus, we can show that:
225 = 30/100 * x,
or, x = (225*100)/30 = 750.
Thus, Paul's rent budget is $750, given that his food budget of $225, is 30 percent of his rent budget.
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A company determines that the revenue, in dollars, for selling a particular model of lamp is given by R(x)=x(−20x+1200), where x is the price of each lamp. At which of the following prices will the company’s revenue be $10,000?
The price that will maximize company’s revenue of $10,000 is $50.
How to calculate the price?From the information, the company determines 50that the revenue, in dollars, for selling a particular model of lamp is given by R(x)=x(−20x+1200),
Therefore, this will be:
R(x) = -20x² + 1200x
10000 = -20x² + 1200x
-20x² + 1200x - 10000 = 0
x² - 60x + 500
x² - 50x - 10x + 500
x(x - 50) - 10(x - 50)
(x - 50)(x - 10)
Therefore, x - 50 = 0
x = 0 + 50 = 50
The price is 50.
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What is the domain of the rational function f of x is equal to the quantity x squared minus x minus 12 end quantity over the quantity x cubed minus 4 times x squared minus 4 times x plus 16 end quantity question mark
{x ∈ ℝ| x ≠ –2, 2, 4}
{x ∈ ℝ| x ≠ –2, 2}
{x ∈ ℝ| x ≠ –3, 4}
{x ∈ ℝ| x ≠ –3, –2, 2, 4}
The domain of the rational function is:
{x ∈ ℝ| x ≠ –2, 2}
What is the domain of the rational function?
Here we have the rational function:
[tex]f(x) = \frac{x^2 - x - 12}{x^3 - 4x^2 - 4x + 16}[/tex]
We want to get the domain of that function. First, we can rewrite the numerator and denominator as:
[tex]x^2 -x - 12 = (x + 3)*(x - 4)[/tex]
[tex]x^3 - 4x^2 - 4x + 16 = (x - 4)*(x + 2)*(x - 2)[/tex]
Then we can rewrite the rational function as:
[tex]f(x) = \frac{(x + 3)*(x - 4)}{(x - 4)*(x + 2)*(x - 2)}[/tex]
This can be simplified to:
[tex]f(x) = \frac{(x + 3)}{(x + 2)*(x - 2)}[/tex]
Now, the domain will be the set of all real numbers, minus the values of x that generate problems.
In this case, the values:
x = -2 and x = 2 make the denominator to be zero, and we can't divide by zero, so we conclue that the domain is:
{x ∈ ℝ| x ≠ –2, 2}
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Choose the correct simplification of (4x − 3)(3x2 − 4x − 3).
the expression simplified is:
[tex]12x^3 - 25x^2 + 9[/tex]
How to simplify the given expression?
Here we have the expression:
[tex](4x - 3)*(3x^2 - 4x - 3)[/tex]
We just need to distribute the product, we will get:
[tex](4x - 3)*(3x^2 - 4x - 3)\\\\(4x)*(3x^2 - 4x - 3) - 3*(3x^2 - 4x - 3)\\\\12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9[/tex]
Now we just need to group terms with the same exponent of x:
[tex]12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9\\\\12x^3 + (-16 - 9)x^2 + (-12 + 12)x + 9\\\\12x^3 - 25x^2 + 9[/tex]
That is the expression simplified
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Melissa Costouras obtains a $3,000 loan for darkroom equipment. She makes six monthly payments of $511.18. Determine the APR.
Using the simple interest formula, it is found that the APR for the loan is of 4.472%.
What is the simple interest formula and when it is used?Simple interest is used when there is a single compounding per time period.
The amount of money after t years in is modeled by:
[tex]A(t) = A(0)(1 + rt)[/tex]
In which:
A(0) is the initial amount.r is the interest rate, as a decimal.The parameters for this problem are:
A(t) = 6 x 511.18 = 3067.08, A(0) = 3000, t = 0.5.
We solve the equation for r to find the APR.
[tex]A(t) = A(0)(1 + rt)[/tex]
[tex]3067.08 = 3000(1 + 0.5r)[/tex]
[tex]1 + 0.5r = \frac{3067.08}{3000}[/tex]
1 + 0.5r = 1.02236
r = (1.02236 - 1)/0.5
r = 0.04472.
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1. Determine the measure of angle B.
B
30
A
64
51
Answer: 51.4°
Step-by-step explanation:
Use law of cosines:
(because you are trying to find angle B, you need to use Cos B)
Cos B = c² + a²-b²/ 2ca
Then, plug in your numbers:
Cos B = 30²+64²-51²/ 2(30)(64)
Simplify:
Cos B = 0.6237
Next, to get rid of Cos B, so that we have just B, you need to do the Arccosine or 0.6237:
B = arccosine (0.6237)
Which gets you: 51.4°
The measure of angle B is 51.41 degrees after applying the cos law of the triangle.
What is the triangle?In terms of geometry, the triangle is a three-sided polygon with three edges and three vertices. The triangle's interior angles add up to 180°.
We have given a triangle in the figure:
To find the measure of the angle B
We can apply cos law:
c² = a² + b² - 2ab cosB
The a, b, and c represents the side lengths of the triangle and the measures are:
a = 30
b = 64
c = 51
51² = 30² + 64² - 2(30)(64) cosB
3840CosB = 4996 - 2601
3840CosB = 2395
CosB = 2395/3840
CosB = 0.623
B = Cos⁻¹(0.623)
B = 51.41 degrees
Thus, the measure of angle B is 51.41 degrees after applying the cos law of the triangle.
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Question 1 of 10
If f(x)=x-1, which of the following is the inverse of f(x)?
O A. f¹(x) = x
OB. f¹(x)=x+1
O C. f¹(x)=x-1
O D. f¹(x) = 1 - x
SUBMIT
What is the area of an equilateral triangle having side 'a' units?
Answer:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]
Step-by-step explanation:
Since an equilateral has all of the sides equal, we can find the height of triangle using: [tex]a^2+b^2=c^2[/tex]. I attached a diagram which should explain how I got the dimensions of the three sides. Using the information from the diagram we get the equation:
[tex]h^2+(\frac{a}{2})^2=a^2[/tex]
Subtract a^2 from both sides
[tex]h^2=a^2-(\frac{a}{2})^2[/tex]
Take the square root of both sides
[tex]h = \sqrt{a^2-(\frac{a}{2})^2}[/tex]
If you know the area of a triangle, it's: [tex]\frac{1}{2}bh[/tex]. In this case the base=a, and the height is what we defined above. Using this we get:
[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a}{2})^2}[/tex]
We can distribute the exponent over the division to get:
[tex]A = \frac{a}{2}*\sqrt{a^2-(\frac{a^2}{4})[/tex]
Now we can rewrite a^2 as 4a^2/4
[tex]A = \frac{a}{2}*\sqrt{\frac{4a^2}{4}-(\frac{a^2}{4})[/tex]
Now add the two fractions:
[tex]A = \frac{a}{2}*\sqrt{\frac{3a^2}{4}[/tex]
We can distribute the square root the division just like how we distributed the exponent 2, since the square root can be expressed as an exponent (1/2)
[tex]A = \frac{a}{2}*\frac{\sqrt{3a^2}}{\sqrt{4}}[/tex]
There's a radical identity that states: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{a*b}[/tex]. We can use this to rewrite one radical as multiple radicals to simplify it:
[tex]A = \frac{a}{2}*\frac{\sqrt{a^2}*\sqrt{3}}{2}[/tex]
Simplify:
[tex]A = \frac{a}{2}*\frac{a*\sqrt{3}}{2}[/tex]
Now multiply the two fractions
[tex]A = \frac{a^2*\sqrt{3}}{4}[/tex]
This is the formula for the area of an equilateral triangle, but it is also often written as:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]
1. In a class of 60 students, a survey was conducted, 30 students had applied for Addis Ababa University, 25 students applied for Bahir Dar University and 24 students applied for Wachemo University. 11 students applied for both Addis Ababa and Bahir Dar Universities, 6 applied for both Addis Ababa and Wachemo Universities, 9 applied for both Wachemo and Bahir Dar Universities while 4 applied neither of the aforementioned universities. Find 1.. number of students that applied for all the universities. 2 number of students that applied for at least two of the universities. 3 number of students that applied at most two universities. 4 number of students that applied for Addis Ababa but not Bahir Dar University? Please I need your help?
1. Consult the linked question. [tex]n(A\cap B\cap W) = \boxed{3}[/tex].
2. We have
[tex]n(A\cap B) = n(A\cap B \cap W) + n(A\cap B \cap W') \\\\ \implies 11 = 3 + n(A\cap B \cap W') \\\\ \implies n(A\cap B\cap W') = 8[/tex]
Similarly we can find
[tex]n(A\cap B' \cap W) = 3[/tex]
[tex]n(A'\cap B\cap W) = 6[/tex]
[tex]n(A\cap B'\cap W') = 16[/tex]
[tex]n(A'\cap B\cap W') = 8[/tex]
[tex]n(A'\cap B'\cap W) = 12[/tex]
Then the total number of students that applied to at least two of the universities is
[tex]\underbrace{n(A\cap B\cap W') + n(A\cap B'\cap W) + n(A'\cap B\cap W)}_{\text{only 2}} + \underbrace{n(A\cap B\cap W)}_{\text{all 3}} = \boxed{20}[/tex]
3. There's a small ambiguity here. Are we interested in students that applied to zero universities? If so, there are
[tex]\underbrace{n(U\setminus(A\cup B\cup W))}_{\text{none}} + \underbrace{n(A\cap B'\cap W') + n(A'\cap B\cap W') + n(A'\cap B'\cap W)}_{\text{only 1}} = \boxed{40}[/tex]
If we want students that applied to at least 1 school, we omit the first term and get a total of 36.
4. Split this set of students into those that also applied to Wachemo and those that did not.
[tex]n(A \cap B') = n(A\cap B' \cap W) + n(A\cap B'\cap W') = \boxed{19}[/tex]
23 gram put into milligrams
Answer:
Step-by-step explanation: 23000 milligrams
Find the slope of the line passing through the points (-4, -9) and (5, -9).
The slope of the line passing through the points (-4, -9) and (5, -9) is 0
How to determine the slope?The points are (-4, -9) and (5, -9).
The slope is calculated using:
Slope = (y2 - y1)/(x2 - x1)
So, we have:
Slope = (-9 + 9)/(5 + 4)
Evaluate
Slope = 0
Hence, the slope of the line passing through the points (-4, -9) and (5, -9) is 0
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Which of the following is the largest Y value from the solution set of the giving system round your answer to the nearest hundredth
The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)
What is the solution of a non-linear system
In this question we have a system of non-linear equation formed by a radical equation and a quadratic equation, which can be solved both analytically and graphically. By equalizing the two formulas we have the following equation:
√(2 · x + 4) = x² - 5 · x + 3
√(2 · x + 4) + 13 / 4 = x² - 2 · (5 / 2) · x + 25 / 4
√(2 · x + 4) + 13 / 4 = (x - 5 / 2) ²
√(2 · x + 4) = (x - 5 / 2) ² - 13 / 4
Then, we square both sides to eliminate the radical sign:
2 · x + 4 = (x - 5 / 2)⁴ - (13 / 2) · (x - 5 / 2)² + 169 / 16
2 · x + 4 = x⁴ + 4 · x³ · (- 5 / 2) + 6 · x² · (- 5 / 2)² + 4 · x · (- 5 / 2)³ + (- 5 / 2)⁴ - (13 / 2) · [x² + 2 · (- 5 / 2) · x + ( - 5 / 2)²] + 169 / 16
2 · x + 4 = x⁴ - 10 · x³ + (75 / 2) · x² - (125 / 2) · x + 625 /16 - (13 / 2) · x² + (65 / 2) · x - 325 / 8 + 169 / 16
2 · x + 4 = x⁴ - 10 · x³ + 31 · x² - 30 · x + 9
x⁴ - 10 · x³ + 31 · x² - 32 · x + 5 = 0
The real roots of this polynomial are:
x₁ ≈ 5.152, x₂ ≈ 2.862, x₃ ≈ 1.797, x₄ ≈ 0.189
Now we evaluate f(x) at each root:
y₁ ≈ 3.782, y₂ ≈ 3.118, y₃ ≈ 2.756, y₄ ≈ 2.092
The largest y-value from the solution set of the nonlinear system is 3.78. (Correct choice: C)
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if sinA=4/5 solve sin2A, cos2A and tan2A
Step-by-step explanation:
1) if m(∠A)∈[0;90°), then
[tex]cos(A)=\sqrt{1-sin^2A} =\frac{3}{5};[/tex]
[tex]sin2A=2sinAcosA=2*\frac{3}{5} *\frac{4}{5}=\frac{24}{25};[/tex]
[tex]cos2A=cos^2A-sin^2A=\frac{9}{25}-\frac{16}{25}=-\frac{7}{25};[/tex]
[tex]tan2A=\frac{sin2A}{cos2A}=-\frac{\frac{24}{25}}{\frac{7}{25}}=-\frac{24}{7}.[/tex]
2) if m∠A∈[90°;180°), then
cos(A)=-0.6;
sin2A=-0.96;
cos2A=-0.28;
tan2A=-24/7.
Jakobe runs a coffee cart where he sells coffee for $1.50 tea for $2 , and donuts for $0.75 . On Monday, he sold 320 items and made $415 . He sold 3 times as much coffee as tea. How many donuts did he sell?
Jakobe sold 150 coffee, 50 tea and 120 donuts at the coffee cart on Monday.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the amount of coffee, y represent the amount of tea and z represent the amount of donuts, hence:
x + y + z = 320 (1)
Also:
1.5x + 2y + 0.75z = 415 (2)
And:
x = 3y (3)
From the equations:
x = 150, y= 50 and z = 120
Jakobe sold 150 coffee, 50 tea and 120 donuts at the coffee cart on Monday.
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Match each function with the corresponding function formula when h(x) = 5 - 3x and g(x) = -3x + 5.
Answer: -5=x
Step-by-step explanation:
kmbtvonpvnp4tnv
solve this pleas i need urgent answer
Answer:
f(n)={n/2 = if n is even= n={2/2
{3n+1 {3(2)+1 =7
Step-by-step explanation:
if n is even u can put any even no in that place and u can get odd no or even no.(i think)
In science class, the average score on a lab report is 87 points, with all students scoring within 1.8 points of the average. If x represents a student's score, write an equation that represents the
minimum and maximum scores.
The equations which represents the minimum and maximum scores of the students is; x = 87 ± 1.8.
Which equations represents the minimum and maximum possible scores of each student?According to the task content, the
Average score of the students in the science class is; 87 points
All students are within 1.8 points of the average,
it therefore follows from the task content that the
minimum score is;
= 87 -1.8
= 85.2
maximum score is;
= 87+1.8
= 88.8 points.
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Answer:
Step-by-step explanation:
A. |x − 87| = 1.8
-hope this helps.
a two digit number has twice as many ones as tens twice the original number is 9 more than the reversed number find the original number
Answer: 36
Step-by-step explanation:
As this is a two-digit number, it will have a digit in its 10's place and a digit in its 1's place. We can represent the 10's digit using x and the 1's digit using y.
Since the ones is twice the number of tens, this means that y is equal to 2*x. Let's put this into an equation.
[tex]y=2x[/tex]
If the 10's digit is x and the 1's digit is y, the original number would be [tex]10x + y[/tex], as x must be multiplied by 10 to get it to the 10's place. Similarly, the reversed number would be [tex]10y + x[/tex], as y must be multiplied by 10 to get it to the 10's place.
Twice the original number (10x+y) is equal to 9 more than (i.e. 9 plus) the reversed number. We can put this into an equation to help us answer the question.
[tex]2(10x+y)=9+10y+x\\20x+2y=9+10y+x\\19x+2y=9+10y\\19x=9+8y[/tex]
Now we have a system of two equations.
[tex]y=2x\\19x=9+8y[/tex]
We can solve this system by substituting y for 2x in the second equation. Then, we can isolate and get the value of x.
[tex]19x=9+8(2x)\\19x=9+16x\\3x=9\\x=3[/tex]
Now that we have the value of x, let's put it back into the first equation and solve for y.
[tex]y=2(3)\\y=6[/tex]
Remember that x is the 10's digit and y is the one's digit of our answer. Since x is 3 and y is 6, our answer is 36.
CheckingWe can quickly check if our answer is right by making sure both conditions in the question are met.
6 (the one's digit) is twice the value of 3 (the ten's digit), making the first condition true. Twice of 36 is 72, which is 9 more than the reversed number, which is 63.
Suppose f′(6)=5 and g′(6)=7.
Find h′(6) where h(x)=4f(x)+5g(x)+1.
The value of the derivative of functions h'(6) as requested in the task content is; 55.
What is the value of h'(6)?Since it follows from the task content that the function h(x)=4f(x)+5g(x)+1.
Hence, the derivative of h(x) can be evaluated as;
h'(x)=4f'(x)+5g'(x)
On this note, by substitution, it follows that;
h'(6)=4(5)+5(7)
h'(6) = 55.
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Choose the algebraic description that maps the image ΔABC onto ΔA′B′C′.
The algebraic description that maps the image ΔABC onto ΔA′B′C′ is (x, y) ⇒ (x + 7, y - 4)
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, rotation, translation and dilation.
Translation is the movement of a point either up, left, right or down in the coordinate plane.
The algebraic description that maps the image ΔABC onto ΔA′B′C′ is (x, y) ⇒ (x + 7, y - 4)
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Mal works at a photo gallery. He charges $50 for a large photo and $40 for a large frame. Sales tax is 5%. How much total tax will a customer pay on both?Answer the questions to show how to write and simplify expressions that represent the problem.
2. Use the distributive property to expand 0.05(50 + 40). (2 points)
Answer:
$4.50
Step-by-step explanation:
We can use the distributive property to solve this problem. 0.05(50+40) is the total sales tax. 0.05 being the percentage and 50 and 40 being the things we need to distribute 0.05 to. This becomes 2.5+2. This is 4.5.
Answer:
$0.45
Step-by-step explanation:
0.05(50 + 40) = 50 x 0.05 + 40 x 0.05 = 0.25 + 0.20 = 0.45
To graph the inequality t>3, you would put an open circle on 3 and shade to the left
Answer:
It is an open circle and you would shade to the right on a number line.
Step-by-step explanation:
The circle is open because you are not including 3. Put numbers in for t and you will see that the numbers greater than 3 are all on the right side of the number line.
-4
-2
4
-2-
0
3
On which interval are both functions increasing?
2
-8
p
0
-X
-2
g(x) = -x - 3| +4
(,0)
Answer: [tex](3, \infty)[/tex]
Step-by-step explanation:
q(x) is increasing for [tex]x > 3[/tex].
p(x) is increasing for [tex]x \geq -2[/tex].
So, both functions are increasing for [tex](3, \infty)[/tex].
What is the value of x
83
58
50
70.5
Answer:
58
Step-by-step explanation:
[tex]x = \frac{141 - 25}{2} = 58[/tex]
Which of the following inequalities matches the graph?
-8 -0
O x>4
O x<4
O y> 4
O y<4
1
Answer:
it matches with equation c
Write the positive or negative number that best
represents the given information.
12 ft above sea level
Answer:
12
Step-by-step explanation:
Usually, above sea level is represented by a positive number and below sea level by a negative number.
12 ft above sea level = 12
I need help to this it would be awesome if you could help.
In the given diagram, the value of JI is 6
Solving equations
From the question, we are to determine JI
From the given diagram, we can write that
JI + IH = x + 22
Also,
JI = 2x + 28
and
IH = 5
Then, we can write that
2x + 28 + 5 = x + 22
Collect like terms
2x - x = 22 -28 -5
x = -11
Now, substitute the value of x into the equation
JI = 2x + 28
JI = 2(-11) + 28
JI = -22 +28
JI = 6
Hence, the value of JI is 6
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what is the ration between 28000 and 14?
The ratio between 28,000 and 14 is 2000 : 1
How to determine the ratio?The numbers are given as:
28,000 and 14
Express as ratio
Ratio = 28000 : 14
Divide each number by 14
Ratio = 2000 : 1
Hence, the ratio between 28,000 and 14 is 2000 : 1
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What is the angle of rotation of the following figure?
This snow flake-like figure can be generated by rotating an end 60° five times around the center of the hexagon. There are two forms: (i) clockwise, (ii) counterclockwise.
What is the angle of rotation of a snow flake?
Geometrically speaking, snow flakes represent regular hexagons. Regular hexagons can divided into six concentric regular triangles, whose central angles have a measure of 60°. This fractal figure can be generated by rotating 60° five times around the center.
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A cube with an edge of 10cm is filled with water.How much can it contain?
The cube can contain 1000 cm^3 of water
How to determine the amount of water?The side length is given as:
L = 10 cm
The amount of water it can contain is calculated as;
V =L^3
So, we have:
V = 10^3
Evaluate
V = 1000
Hence, the cube can contain 1000 cm^3 of water
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the cowboys football team started at the zero yard line.throughout the course of the first quarter the team gained 7 yards, lost 8 yards and then lost 7 more yard. what is the total of these three plays? how many yards did the team gain or lose?
The net number of yards lost by the cowboys football team is; 8 yards.
How many yards did the team gain or lose?It follows from the task content that the number of yards gained or lost by the football team during each of the three plays is; gained 7 yards, lost 8 yards and then lost 7 more yard.
Hence, by polarising yard gain as positive and yard loss as negative; we have;
Net total = 7 - 8 -7 = -8
Hence, the team lost 8 yards in the total game.
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