If both a and b are positive numbers and (b)/(a) is greater than 1, then a-b will be negative.
This is because when (b)/(a) is greater than 1, it means that b is greater than a. So when you subtract a from b, you will get a negative number.
For example, let's say a = 2 and b = 5.
(b)/(a) = (5)/(2) = 2.5, which is greater than 1.
So when we subtract a from b, we get:
b - a = 5 - 2 = 3, which is a positive number.
But when we subtract b from a, we get:
a - b = 2 - 5 = -3, which is a negative number.
Therefore, if both a and b are positive numbers and (b)/(a) is greater than 1, then a-b will be negative.
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9. Simplify each expression: a. √(1 + cos 76° / 2) b. (sin 158.2°) / (1 + cos 158.2°) 10. Verify that each equation is an identity. a. sin 2x / 2 sin x = cos^2 x/2 – sin^2 x/2 b. tan θ/2 = csc θ – cot θ
LHS ≠ RHS and the equation is not an identity.
9. Simplify each expression:
a. √(1 + cos 76° / 2)
b. (sin 158.2°) / (1 + cos 158.2°)
10. Verify that each equation is an identity.
a. sin 2x / 2 sin x = cos^2 x/2 – sin^2 x/2
b. tan θ/2 = csc θ – cot θ
Answer:
9. a. √(1 + cos 76° / 2) = √(1 + 0.2419 / 2) = √(1 + 0.12095) = √(1.12095) = 1.0589
b. (sin 158.2°) / (1 + cos 158.2°) = (0.12088) / (1 + (-0.9927)) = 0.12088 / 0.0073 = 16.566
10. a. sin 2x / 2 sin x = cos^2 x/2 – sin^2 x/2
LHS = sin 2x / 2 sin x = 2 sin x cos x / 2 sin x = cos x
RHS = cos^2 x/2 – sin^2 x/2 = (cos x + sin x)(cos x - sin x) / 4 = (1)(cos x - sin x) / 4 = cos x - sin x / 4
Therefore, LHS ≠ RHS and the equation is not an identity.
b. tan θ/2 = csc θ – cot θ
LHS = tan θ/2 = sin θ/2 / cos θ/2
RHS = csc θ – cot θ = 1 / sin θ - 1 / cos θ = (cos θ - sin θ) / (sin θ cos θ)
Therefore, LHS ≠ RHS and the equation is not an identity.
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HELP ME PLS ANYBODY
ITS DUE TODAY AND I NEED HELP ASAP
Answer:
1. around 4.86 or 4.9 for the nearest tenth
2. 17 months
Step-by-step explanation:
1. (1.7×10^6)/(3.5×10^5) = (1.7/3.5)×(10^6/10^5) = 0.4857×10 = 4.857
Therefore, 1.7×10^6 is about 4.857 times as great as 3.5×10^5.
2. start by finding out how much Erica still owes after the down payment:
Total cost - Down payment = $1,867 - $320 = $1,547
divide the amount still owed by the monthly payment to find out how many months Erica will be paying:
$1,547 ÷ $91 per month = 17 months (rounded up)
Therefore, Erica will be paying for the bike for 17 months
Before a computer programmer optimized a program, 450 clock cycles were required to process an input file. After optimization, the file was processed in 60 clock cycles. To the nearest percent, what
The nearest percent, the percent decrease in clock cycles after optimization is approximately 87%.
What is optimization?Optimization is the process of improving the performance or efficiency of a system, process, or algorithm by finding the best possible solution within a given set of constraints or limitations.
The percent decrease in clock cycles after optimization can be calculated using the formula:
Percent decrease = [(original value - new value) / original value] x 100%
where the original value is the number of clock cycles required before optimization, and the new value is the number of clock cycles required after optimization.
Using the given values, we have:
Percent decrease = [(450 - 60) / 450] x 100%
= (390 / 450) x 100%
= 86.67%
Therefore, to the nearest percent, the percent decrease in clock cycles after optimization is approximately 87%.
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There has been a recent outbreak of a deadly disease in a particular locality. The local health council has determined that the chance of survival for a person who tests positive for the disease is 30%. In one of the public hospitals in the said locality, 12 people have been admitted and tested positive for the disease. Of these 12 individuals:
Of these 12 individuals: Expected number of survivors ≈ 4.
Based on the given information, the chance of survival for the 12 people admitted to the hospital who tested positive for the disease is 30%.
If the chance of survival for a person who tests positive for the disease is 30%, and 12 people have been admitted and tested positive for the disease, then the expected number of survivors is 30% of 12, which is 3.6.
However, since it is not possible for 0.6 of a person to survive, we can round this number to the nearest whole number, which is 4.
Therefore, we can expect that 4 out of the 12 people who tested positive for the disease will survive. This can be written as:
Expected number of survivors = (Chance of survival) × (Number of people who tested positive) = (0.30) × (12) = 3.6 ≈ 4
So, the answer is 4.
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what is 7.5cm = to in mm?
Answer:
75mm
Step-by-step explanation:
1cm = 10mm
u multiply 7.5 by 10
Answer:75
Step-by-step explanation:
A caterer made 24 sandwiches for a company lunch. When they arrived at the location. they discovered the guest list was more than they had anticipated. They decided to cut the sandwiches into ( l)/(3) size pleces. How many ( 1)/(3) size pieces were they able to make?
The caterer was able to make 72 (1)/(3) size pieces from the 24 sandwiches.
To find this answer, we can use the following equation:
24 sandwiches × 3 (1)/(3) size pieces per sandwich = 72 (1)/(3) size pieces
This equation tells us that for every sandwich, the caterer was able to make 3 (1)/(3) size pieces. By multiplying the number of sandwiches by the number of (1)/(3) size pieces per sandwich, we can find the total number of (1)/(3) size pieces the caterer was able to make.
Therefore, the caterer was able to make 72 (1)/(3) size pieces from the 24 sandwiches.
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Consider the decision sapling in which you can choose either a sure thing worth $5M or a gamble that has probability of 0.4 of giving a return of $2M; otherwise the return on the gamble will be $7M. Suppose we can purchase a forecast for the gamble, where the forecast has probability p of being accurate. (So the false positive and false negative rates are both 1-p). a. What is the EMV of the gamble given the forecast is that the return will be $7M? If it is $2M? b. If the forecast is $7M, for what range of p will it be optimal to choose the gamble? c. Give an expression for the EVII as a function of p, and graph it.
The EMV (Expected Monetary Value) of the gamble is calculated as follows:
EMV = (probability of return) x (value of return)
a. If the forecast is that the return will be $7M, then the EMV of the gamble is:
EMV = (0.4) x ($7M) = $2.8M
If the forecast is that the return will be $2M, then the EMV of the gamble is:
EMV = (0.6) x ($2M) = $1.2M
b. If the forecast is $7M, the range of p for which it will be optimal to choose the gamble is when the EMV of the gamble is greater than the sure thing of $5M. This can be calculated by setting the EMV of the gamble equal to $5M and solving for p:
$5M = (0.4) x ($7M) + (1-p) x ($2M)
$5M = $2.8M + $2M - $2Mp
$2.2M = $2Mp
p = 1.1
Therefore, the range of p for which it will be optimal to choose the gamble is when p > 1.1.
c. The EVII (Expected Value of Imperfect Information) is the difference between the EMV of the gamble with the forecast and the EMV of the gamble without the forecast. This can be expressed as a function of p as follows:
EVII = (p x EMV with forecast) - (1-p) x EMV without forecast
EVII = (p x ($2.8M)) - (1-p) x ($1.2M)
EVII = $2.8Mp - $1.2M + $1.2Mp
EVII = $4Mp - $1.2M
To graph this function, plot p on the x-axis and EVII on the y-axis. The slope of the line will be $4M and the y-intercept will be -$1.2M.
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DBA QUESTION #4
How would you identify a perfect square trinomial?
Give an example by identifying a perfect square trinomial and then simplifying it.
Answer:
A perfect square trinomial is a trinomial expression of the form:
a^2 + 2ab + b^2
Where a and b are constants, it can also be written as (a + b)^2.
To identify a perfect square trinomial, we can check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms.
For example, let's consider the expression:
x^2 + 4x + 4
The first term is x^2, which is a perfect square. The last term is 4, which is also a perfect square. The middle term is 4x, twice the product of the square roots of x^2 and 4 (i.e., 2x). Therefore, this expression is a perfect square trinomial:
x^2 + 4x + 4 = (x + 2)^2
To simplify this expression, we can use the fact that (a + b)^2 = a^2 + 2ab + b^2:
(x + 2)^2 = x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4
Therefore, the perfect square trinomial x^2 + 4x + 4 is equivalent to (x + 2)^2.
Step-by-step explanation:
Two cars travel at the same speed to different destinations. Car A reaches its destination in 12 minutes. Car B reaches its destination in 18 minutes. Car B travels 4 miles farther than Car A. How fast do the cars travel? Write your answer as a fraction in simplest form.
Answer:
chatGPT
Step-by-step explanation:
Let's denote the speed of each car as v, and the distance that Car A travels as d. Then we can set up two equations based on the information given:
d = v * (12/60) (since Car A reaches its destination in 12 minutes)
d + 4 = v * (18/60) (since Car B travels 4 miles farther than Car A and reaches its destination in 18 minutes)
Simplifying the equations by multiplying both sides by 60 (to convert the minutes to hours) and canceling out v, we get:
12v = 60d
18v = 60d + 240
Subtracting the first equation from the second, we get:
6v = 240
Therefore:
v = 240/6 = 40
So the cars travel at a speed of 40 miles per hour.
Please help me did my homework for math
Answer:
1 = 128
2 = 52
3 = 52
4 = 128
5 = 128
6 = 52
7 = 52
8 = 128
Find the Value of "h" h² +4h+111=0
Answer: h=−2+i√107,−2−i√107
The variable a is jointly proportional to the cube of b and the square of c. If a=433 when b=5 and c=7, what is the value of a when b=4 and c=4? Round your answer to two decimal places if necessary.
The value of a when b=4 and c=4 is 72.70. Rounded to two decimal places, the answer is 72.70.
The variable a is jointly proportional to the cube of b and the square of c. This means that the relationship between a, b, and c can be expressed as: a = k * b^3 * c^2, where k is a constant of proportionality.
When a=433, b=5, and c=7, we can plug these values into the equation to find the value of k:
433 = k * 5^3 * 7^2
433 = k * 125 * 49
433 = k * 6125
k = 433/6125
k = 0.07069
Now, we can use this value of k to find the value of a when b=4 and c=4:
a = k * b^3 * c^2
a = 0.07069 * 4^3 * 4^2
a = 0.07069 * 64 * 16
a = 72.70
Therefore, the value of a when b=4 and c=4 is 72.70. Rounded to two decimal places is 72.70.
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a Right circular cylinder has the dimensions shown below
R= 17.2 m
H = 15.3
what is the volume of the cylinder
The volume of the right circular cylinder is approximately 14,864.77 cubic meters.
What is the right circular cylinder?
A right circular cylinder is a three-dimensional solid shape that consists of a circular base and a curved side that is perpendicular to the base. The term "right" refers to the fact that the axis of the cylinder is perpendicular to the base, and the term "circular" refers to the shape of the base, which is a circle.
The formula for the volume of a right circular cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder. The formula for the surface area of a right circular cylinder is A = 2πrh + 2πr^2, where r is the radius of the base and h is the height of the cylinder.
Right circular cylinders are commonly used in engineering, architecture, and everyday objects such as cans, pipes, and containers.
The formula for the volume of a right circular cylinder is [tex]V = \pi r^2h[/tex], where r is the radius of the base and h is the height of the cylinder.
In this case, the radius is 17.2 meters and the height is 15.3 meters.
Plugging in these values, we get:
[tex]V = \pi (17.2)^2(15.3)[/tex]
V ≈ 14,864.77 cubic meters
Therefore, the volume of the right circular cylinder is approximately 14,864.77 cubic meters.
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What is the multiplicity of the zero x=1 for the function f(x)=x^(2)(x-1)^(4)(x+5)
The multiplicity of the zero [tex]x=1[/tex] for the function [tex]f(x)=x^2(x-1)^(4)(x+5)[/tex] is 4.
In a polynomial function, the multiplicity of a zero is the number of times that zero appears as a factor in the function. In other words, it is the exponent of the factor corresponding to that zero.
In the given function [tex]f(x)=x^2(x-1)^4(x+5)[/tex], we can see that the zero x=1 corresponds to the factor (x-1)^(4). This means that the multiplicity of the zero x=1 is 4, as it appears as a factor 4 times in the function.
Therefore, the multiplicity of the zero [tex]x=1[/tex] for the function [tex]f(x)=x^2(x-1)^(4)(x+5)[/tex] is 4.
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Dividing a polynomial by a monomial: (-15xz+18x^(6)z^(7)+25x^(5)z^(2))-:(-3x^(5)z^(2)) mplify your answer as much as possible.
The simplest form obtained by the given division is (-25 - 18x^(5)z^(6) + 15x^(4)z)/(3x^(5)z^(2)) .
To divide a polynomial by a monomial, we simply divide each term in the polynomial by the monomial. So, we will divide each term in (-15xz+18x^(6)z^(7)+25x^(5)z^(2)) by (-3x^(5)z^(2)).
First, we will divide -15xz by -3x^(5)z^(2):
-15xz/(-3x^(5)z^(2)) = 5x^(-4)z^(-1)
Next, we will divide 18x^(6)z^(7) by -3x^(5)z^(2):
18x^(6)z^(7)/(-3x^(5)z^(2)) = -6x^(1)z^(5)
Finally, we will divide 25x^(5)z^(2) by -3x^(5)z^(2):
25x^(5)z^(2)/(-3x^(5)z^(2)) = -25/3
So, our final answer is:
5x^(-4)z^(-1) - 6x^(1)z^(5) - 25/3
In simplified form, this is:
(-25 - 18x^(5)z^(6) + 15x^(4)z)/(3x^(5)z^(2))
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At summer camp the counselors want to order beads for crafts There are 597 campers at the camp each bead project uses 89 beads about how many beads should the counselors order
Answer:
597 x 89 = 53133.
53133 beads
Step-by-step explanation:
If possible, simplify the following expression. Otherwise, use the "Simplified" button. (15x^(2)+13x+2)/(3x-2) where x!=(2)/(3)
The final simplified expression is (3x+1)(5x+2)/(3x-2).
The given expression is (15x^(2)+13x+2)/(3x-2). We can try to simplify this expression by factoring the numerator and denominator, and then canceling out any common factors.
First, let's factor the numerator:
15x^(2)+13x+2 = (3x+1)(5x+2)
Now, let's factor the denominator:
3x-2 = (3x-2)
There are no common factors between the numerator and denominator, so we cannot simplify the expression any further.
Therefore, the simplified expression is:
(15x^(2)+13x+2)/(3x-2) = (3x+1)(5x+2)/(3x-2)
Since x!=(2)/(3), we do not need to worry about any undefined values.
So, the final simplified expression is:
(3x+1)(5x+2)/(3x-2)
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unit linear relationships student handout 1 answer key
In statistics, a linear relationship is a connection that runs in a straight line between two variables (or linear association).
What do you mean by linear relationships?Both a mathematical representation of a linear relationship, in which the dependent variable is produced by multiplying the independent variable by the slope coefficient and a constant, and a graphic representation of a linear relationship, in which the variable and constant are connected by a straight line.
A polynomial or non-linear (curved) relationship can be contrasted with a linear relationship.
A linear relationship is one that mathematically answers the question:
y = mx + b
Where,
m = slope
b = y-intercept
"X" and "Y" are two variables in this equation.
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each interior angle of a regular polygon measures 156 how many sides does the regular polygon have ?
Answer: 15
Step-by-step explanation:
Answer:
[tex]\boxed{n= 15}[/tex]
Step-by-step explanation:
we can use the following formula:
[tex]\alpha = \frac{180(n-2)}{n}[/tex]
This formula helps to calculate the sum of the interior angles of a polygon, where:
[tex]\alpha[/tex] = interior angle[tex]n[/tex] = number of sideswe have the value of the interior angle, and we need "n", so we will solve for "n":
[tex]156= \frac{180(n-2)}{n}\\156n=\frac{180(n-2) \not{n}}{\not{n}}\\156n= 180n-360\\-24n=-360\\n= 15[/tex]
With this we have solved the exercise.
[tex]\text{-B$\mathfrak{randon}$VN}[/tex]
Identify the similar triangles.
E 6 H
F
order of size.
AFEH
Identify the similar triangles in increasing
tenth.
X =
X
~ A
G
Then find the value of x to the nearest
24
The similar triangles in increasing order are ΔEHF ~ ΔFHG ~ ΔEFG
The value of x = 13.5
What are Similar Triangles?Two triangles are similar if and only if:
All their corresponding angles are congruent.The ratios of their corresponding sides are equal.They have the same shape but not necessarily the same sizeTherefore, from the diagram, we can conclude that the similar triangles in increasing order of size is given as:
ΔEHF ~ ΔFHG ~ ΔEFG
Using the altitude rule, we will have the following proportion:
6/9 = 9/x
6x = 81
x = 13.5
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Find all the missing angels
Answer:
15, 15
Step-by-step explanation:
Since this triangle is a right triangle has it's other two angles at 45 degrees, (180-90-45=45 for the last angle) we know that it's a 45-45-90 triangle and can use the rules in the image below to solve the other two sides:
Because the hypotenuse of this triangle is 15[tex]\sqrt{2}[/tex], then we know that the other two side lengths are 15[tex]\sqrt{2}[/tex]/[tex]\sqrt{2}[/tex], or in this case, just 15.
Answer: 15, 15
Step-by-step explanation:
It is a 45, 45, 90 triangle. This means that both of the legs have the same lengths and the hypotenuse is sqrt2 times the length of one of the legs.
Since sqrt2 is irrational, it is easy to look at the coefficient of it. Since the coefficient is 15, the lengths of the sides are also 15.
I need the domain and the range of this graph! Im reposting this question please help
Based on the graph of this linear function, the domain and range are as follows;
Domain = {0, 100}.
Range = {450, 1200}.
What is a domain?In Mathematics, a domain is the set of all real numbers for which a particular function is defined.
Additionally, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:
Domain = {0, 100}.
Range = {450, 1200}.
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Initially, there were only 3 weeds at a park. The weeds grew at a rate of 5% each week. The following function represents the weekly weed growth: f(x) = 3(1.05)x. Rewrite the function to show how quickly the weeds grow each day and calculate this rate as a percentage.
Answer: To rewrite the function to show how quickly the weeds grow each day, we need to use the fact that there are 7 days in a week. Let's divide both sides of the original function by 7 to get:
f(x/7) = 3(1.05)^(x/7)
This function gives the amount of weed growth after x/7 weeks, which is equivalent to x days. To find the daily weed growth rate, we need to take the derivative of this function with respect to x and evaluate it at x=0:
f'(0) = (d/dx) 3(1.05)^(x/7)
Using the chain rule, we get:
f'(0) = 3ln(1.05)/7 ≈ 0.00797
This means that the weeds grow at a daily rate of approximately 0.797%, or 0.00797 as a decimal.
Step-by-step explanation:
Q1 A line passes through the point(5,10)and(−3,12)find. a. A point-alop equation of this line b. a slop-intercept equatice of this line. Q2 Find the equation of the liae which: 4. puses throogh(68)and purilled io the lise with equationy=2x−6b. pases throogh (12,3) and perpendicular to the line with equationy=c. pasies through the point(1,2)and (5,6) Q3 Letf(x)=3x+5andg(x)=1/(vx−3). Find a.(f+g)(x)b.(f⋅g)(x)c.(2f+3g)(x)d.(3g−4)(x)Q4 Letf(x)=x2andg(x)=vx+1. Find a.(f2g)mb.(∘f min Q5 Deternine tle domsh and the range of the foctowing functens:
f(x)=3x+5
g(x)=1/(x-3)
Q1 The point-slope equation of the line passing through the points (5,10) and (-3,12) is y-10=2(x-5) and the slope-intercept equation of the same line is y=2x-9.
Q2 a) The equation of the line which passes through (8,0) and is parallel to the line with equation y=2x-6 is y=2x.
b) The equation of the line which passes through (12,3) and is perpendicular to the line with equation y=2x-6 is y=-1/2x+9.
c) The equation of the line which passes through the points (1,2) and (5,6) is y=2x-1.
Q3 a) (f+g)(x) = 3x+5 + 1/(x-3)
b) (f⋅g)(x) = 3x+5 * 1/(x-3)
c) (2f+3g)(x) = 6x+10 + 3/(x-3)
d) (3g-4)(x) = 3/(x-3) - 4
Q4 a) (f2g)(x) = (x2)2/(x+1)
b) (∘f min g)(x) = x/(x+1)
Q5 The domain of the function f(x)=3x+5 and g(x)=1/(x-3) is all real numbers except 3 and the range of both the functions is all real numbers.
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A gardener is planting two types of trees: Type A is 5 feet tall and grows at a rate of 4 inches per year. Type B is 2 feet tall and grows at a rate of 10 inches per year. Algebraically determine exactly how many years it will take for these trees to be the same height.
It will take 18 years for the Type A and Type B trees to be the same height.
What is the midpoint formula?A location in the centre of a line connecting two places is referred to as the midpoint. The midpoint of a line is located between the two reference points, which are its ends. The line that connects these two places is split in half equally at the halfway. In addition, the halfway is reached if a line is drawn to divide the line that connects these two places.
Let us suppose the number of years = y.
The height of the tree of type A is thus,
5 feet + (4 inches/year)(y)
= 5 + 4/12 y feet
= 5 + 1/3 y feet
For the type B:
2 feet + (10 inches/year)(y years)
= 2 + 10/12 y feet
= 2 + 5/6 y feet
To get the height of the two trees as equal we have:
5 + 1/3 y = 2 + 5/6 y
3 + 1/3 y = 5/6 y
3 = 1/6 y
18 = y
Hence, it will take 18 years for the Type A and Type B trees to be the same height.
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What is the area of the parallelogram?
37 in.
31 in.
425.5 in.²
713 in. 2
851 in.2
1147 in.²
23 in.
Area of the parallelogram, using the area formula = 851 in².
What is the area of a parallelogram?The area of a parallelogram, which is expressed in square units, is the total number of unit squares that may fit inside of it (like cm², m², in² etc). It is the region that a two-dimensional parallelogram encloses or encompasses.
Hence, the following formula can be used to determine a parallelogram's area:
Area of a parallelogram is equal to b × h square units, where b is the base's length and h is its height.
Here in the question,
b = 37in.
h = 23in.
Area = b × h
= 37 × 23
= 851 in²
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An inequality is shown. -1/3x + 1/2 < 3.5 what is the solution to the inequality?
A. x > -12
B. x < -12
C. x > -9
D. x < -9
The solution to the inequality is option C. x > -9.
What is Linear Inequalities?Linear inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠ and the value of the exponent of the variable is 1.
The given inequality is,
-1/3x + 1/2 < 3.5
We have to find the solution of the inequality.
-1/3 x + 0.5 < 3.5
Subtracting both sides by 0.5, we get,
-1/3 x < 3.5 - 0.5
-1/3 x < 3
Multiplying both sides by -3,
x > -9
Hence the solution is x > -9.
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1
Destiny combines 8.27 liters of red paint with 6.65 liters of blue paint to make purple paint. She
pours the paint equally into 2 containers, and has 1.56 liters of paint left over. How many liters
of paint are in each container? >
Container
?
2
There are
Solve on paper. Then check your work on Zearn.
Total paint = 8.27 + 6.65
M1|L16
3
4
Decimal Problem Solving
5 6
Container
liters of paint in each container.
7 8 90
Paint left
1.56
Enter ✔
The number of liters in each container is 8.24 liters.
What is division?Division is a mathematical operation, in which we distribute the number in equal parts, the number on the upper side is the total quantity and the number on the bottom side is equal parts of numbers which have to be distributed.
We denote division by '÷' this symbol.
Destiny combined a total of 8.27 + 6.65 = 14.92 liters of paint to make purple paint.
She poured this paint equally into 2 containers, which means each container has half of the total amount of paint:
= 14.92 / 2
= 7.46 liters
However, Destiny also had 1.56 liters of paint left over that she didn't use.
To divide the remaining paint equally between the two containers, each container gets half of the remaining paint:
= 1.56 / 2
= 0.78 liters
Therefore, each container has 7.46 + 0.78 = 8.24 liters of paint.
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Find the equation of the curve with gradient function 3x2+5x4 that passes through (1,8)
The equation of the curve that passes through the point (1, 8) and has the gradient function 3x² + 5x⁴ is y = x³ + x⁵ + 6
In this problem, we are given the gradient function 3x² + 5x⁴ and the point (1, 8) through which the curve passes. To find the equation of the curve, we need to integrate the gradient function and add the constant of integration. Let's start by integrating 3x² + 5x⁴:
∫(3x² + 5x⁴)dx = x³ + x⁵ + C
where C is the constant of integration.
Now we have the general equation of the curve, but we need to find the specific value of C that will give us the curve that passes through the point (1, 8). We can do this by plugging in the values of x and y into the equation:
8 = 1³ + 1⁵ + C
8 = 1 + 1 + C
C = 6
Therefore, the equation of the curve is:
y = x³ + x⁵ + 6
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a varies directly as b and inversely as the square of c. If a=113 when b=7 and c=8, find a if b=5 and c=3. Round your answer to two decimal places if necessary.
The value of a is 576.43, when a varies directly as b and inversely as the square of c.
Given that a varies directly as b and inversely as the square of c, we can write the equation as:
a = k * (b/c²)
Where k is the constant of proportionality.
We are given that a=113 when b=7 and c=8, so we can plug these values into the equation and solve for k:
113 = k * (7/8²)
113 = k * (7/64)
k = 113 * (64/7)
k = 1036.57
Now that we know the value of k, we can plug in the new values of b and c to find a:
a = 1036.57 * (5/3²)
a = 1036.57 * (5/9)
a = 576.43
Therefore, when b=5 and c=3, a=576.43.
Round your answer to two decimal places if necessary, so the final answer is:
a = 576.43
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