An open box is formed by cutting squares with side lengths of 3 inches from each corner of a square piece of paper. what is a side length of the original paper if the box has a volume of 675 cubic inches?
-13x=90-2y
-6x=48-2y
The solution to the system of equations is x = 6 and y = 84.
Explanation:The given equations are:
-13x = 90 - 2y
-6x = 48 - 2y
To find the values of x and y, we can solve the system of equations. We can start by multiplying the second equation by -2 to eliminate the y variable. This gives us:
12x = -96 + 4y
Now we have two equations:
-13x = 90 - 2y
12x = -96 + 4y
We can add the two equations together:
-13x + 12x = 90 - 2y + (-96 + 4y)
-x = -6
Then, we can divide both sides of the equation by -1 to solve for x:
x = 6
Substituting the value of x back into one of the original equations, we can solve for y. Let's use the first equation:
-13(6) = 90 - 2y
-78 = 90 - 2y
-2y = -168
y = 84
Therefore, the solution to the system of equations is x = 6 and y = 84.
Find the exact value of the following limit: the limit as x goes to 0 of the quotient of the quantity e raised to the 3 times x power minus 3 times x minus 1 and x squared.
Type answer as a decimal.
The value of the limit is 4.5 .
What is limit?An output value that a function approaches for the specified input values is referred to as a limit.
Calculus and mathematical analysis depend on limits, which are also used to determine integrals, derivatives, and continuity.
The given limit is,
F(x) = [tex]lim_{x- > 0} (e^{3x}-3x-1)/x^2[/tex]
Differentiate with respect to x,
F'(x) = [tex]lim_{x- > 0} d/{dx}(e^{3x}-3x-1)/x^2[/tex]
= [tex]lim_{x- > 0} d/{dx}(e^{3x}-3x-1)/d/dx(x^2)[/tex]
= [tex]lim_{x- > 0}\ 3e^{3x}-3/2x[/tex]
Differentiate again with respect to x,
F''(x) = [tex]lim_{x- > 0}\ d/dx(3e^{3x}-3/2x)[/tex]
= [tex]lim_{x- > 0} (9e^{3x)}/2}[/tex]
= [tex]9 .e^0/2[/tex]
= [tex]9/2[/tex]
= [tex]4.5[/tex]
The limit of the function is 4.5 .
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What are the domain, range, and asymptote of h(x) = (1.4)x + 5?
How many roots do the following equations have?
-12x2 - 25x + 5 + x3 = 0
A. 4
B. 5
C. 3
D. 2
From a group of six people, two individuals are to be selected at random. how many possible selections are there
Final answer:
To find the number of possible selections when choosing two individuals from a group of six people at random, use the combinations formula C(6, 2) = 6! / (2!(6-2)!) which equals 15 possible selections.
Explanation:
From a group of six people, selecting two individuals at random involves calculating the number of combinations. The formula for combinations is C(n, k) = n! / (k!(n-k)!), where 'n' represents the total number of items and 'k' represents the number of items to choose.
In this scenario, with n being 6 (the total number of people) and k being 2 (the number of people to select), the calculation would be:
C(6, 2) = 6! / (2!(6-2)!) = (6 * 5) / (2 * 1) = 15
Therefore, there are 15 possible selections when choosing two individuals from a group of six people at random.
What are the vertex, focus, and directrix of the parabola with the equation x^2+8x+4y+4=0
Answer:
Vertex=(-4,3)
Focus=(-4, 2)
Directrix: y=4
Step-by-step explanation:
Given the equation
[tex]x^2+8x+4y+4=0[/tex]
we have to find the vertex, focus, and directrix of parabola.
[tex]x^2+8x+4y+4=0[/tex]
[tex]y=\frac{-x^2}{4}-2x-1[/tex]
[tex]h=\frac{-b}{2a}=\frac{2}{2(\frac{-1}{4})}=-4[/tex]
Now, k can be calculated by putting x=4 and y=k in given equation
[tex]k=-1-\frac{16}{4}-2(-4)=-1-4+8=3[/tex]
The vertex is (h,k) i.e (-4,3)
[tex]y=\frac{-x^2}{4}-2x-1[/tex]
[tex]y=\frac{1}{4}(-x^2-8x-4)[/tex]
[tex]=\frac{1}{4}(-x^2-8x-16+16-4)[/tex]
[tex]y=\frac{1}{4}(-(x+4)^4+12)[/tex]
[tex]y=\frac{-1}{4}(x+4)^2+3[/tex]
which is required vertex form [tex]4p(y-k)=(x-h)^2[/tex]
gives p=-1
Focus=(-4, 3+(-1))=(-4,2)
Now directrix can be calculated as
y=3-p=3-(-1)=4
house blend coffee is 50% columbian beans and special blend coffee is 80% columbian beans. how much of each should be used to produce 100kg of a blend that is 68% columbian beans
Let us say that,
x = mass of 50% Columbian beans required
y = mass of 80% Columbian beans required
To solve this problem, we set up two mass balance equations:
Overall mass balance: 100 = x + y
x = 100 – y ---> 1
Coffee mass balance: 0.68 (100) = 0.50 x + 0.80 y
68 = 0.50 x + 0.80 y ---> 2
Combining equations 1 and 2:
68 = 0.50 (100 – y) + 0.80 y
68 = 50 – 0.50 y + 0.80 y
18 = 0.30 y
y = 60 kg
Calculating for x using equation 1:
x = 100 – y
x = 100 – 60
x = 40 kg
Answer:
40 kg of 50% Columbian beans required
60 kg of 80% Columbian beans required
The key idea of the transformation called a reflection is _____.
the object moves the plane a certain angle measure
the object moves in a straight line movement
the object and its image have opposite orientations
the rotating of a plane about a fixed point
Answer: The object and its image have opposite orientations
Step-by-step explanation:
A reflection is a type of rigid transformation in which the preimage is flipped across a line of reflection to produce the image, such that the figure and its image have opposite orientations.
Since, it preserves distance.
Therefore, the points of the image have the same distance from the line as the points of pre-image on the opposite side of the line.
Hence, the right option is "the object and its image have opposite orientations".
a culture started with 5000 bacteria. after 2 hours, it grew to 6500 bacteria. Predict how many bacteria will be present after 18 hrs
What is the logarithmic function modeled by the following table
x
8
16
32
f(x)
3
4
5
A. F(x)= log_x2
B. F(x)= log_2 x
C. F(x)= 2 log_10 x
D. F(x)= x log_10 2
Log_2 8 = 3 (because 2^3 = 8), log_2 16 = 4 and log_2 32 = 5.
What are Logarithms?A base must be raised to a certain exponent or power, or logarithm, in order to produce a specific number. If bx = n, then x is the logarithm of n to the base b, which is expressed mathematically as x = logb n.
For instance, 23 = 8; hence, 3 is the base-2 logarithm of 8 or 3 = log2 8. In the same way, 2 = log10 100 since 102 = 100. The latter type of logarithms, those with base 10, are known as common or Briggsian logarithms and are denoted by the letter log n.
Logarithms, which were developed in the 17th century to expedite calculations, significantly decreased the amount of time needed to multiply integers with numerous digits.
Therefore, Log_2 8 = 3 (because 2^3 = 8), log_2 16 = 4 and log_2 32 = 5.
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solve the equation 2x^2-1x=5x
Please enter any square root values using "sqrt" or the decimal answer rounded to the nearest hundredth (i.e. / = sqrt(2) or 1.41).
Answer:
x=0, x=3
Step-by-step explanation:
You unfold chicken wire to make a circular pen with a diameter of 2.9 meters. how many meters of chicken wire do you need?
circumference = PI x D
using 3.14 for PI
2.9 x 3.14 = 9.106 = 9.11 meters of chicken wire are needed.
The stack on the left is made up of 15 pennies, and the stack on the right is also made up of 15 pennies. If the volume of the stack of pennies on the left is 360 mm3, what is the volume of the stack of pennies on the right in cubic millimeters? Use 3.14 for pi. (Hint: only enter numerals in the answer blank)
Answer:
360 is the answer. Just type in 360 in the answer blank.
Step-by-step explanation:
A liquid substance was left at the scene of the crime. Krisann uses a beaker that measures to the nearest tenth of a liter and finds that it is 4.9 liters. If there are five or more liters of the substance, it needs to be sent to the lab for testing. If it is less than five liters, the test can be done immediately. Does Krisann need to send the substance to the lab? Why or why not?
If the volume of the substance of 4.9 liters is from x rounded to the nearest tenth of a liter, that means 4.85 <= x < 4.95 then this does NOT leads to x > 5. Therefore the substance does not need to be sent to the lab.
If BE−→ B E → bisects ∠ABD and m∠ABE = 62°, find m∠ABD
Based on the definition of an angle bisector, m∠ABD = 124°.
What is Angle Bisector?An angle bisector is a line segment that divides an angle into two smaller angles that are congruent.
BE is an angle bisector, therefore:
m∠ABD = 2(m∠ABE)
m∠ABD = 2(62)
m∠ABD = 124°
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Evaluate the function rule for the given value. Y=6*4 for x=-3
A game involves tossing a biased coin that has a 60% probability of landing on heads. If a player wins $50 when heads appears, what is the expected value of a player's winnings?
The product of (4z2 + 7z – 8) and (–z + 3) is –4z3 + xz2 + yz – 24.
What is the value of x?
What is the value of y?
Answer:
[tex]\text{The value of x and y is } x=5, y=29[/tex]
Step-by-step explanation:
Given the product of two expressions
[tex]4z^2+7z-8\text{ and }-z+3\text{ is }-4z^3+xz^2+yz-24[/tex]
we have to find the value of x and y
First we find the product of
[tex]4z^2+7z-8\text{ and }-z+3[/tex]
[tex](4z^2+7z-8)(-z+3)[/tex]
Opening the brackets
[tex]4z^2(-z+3)+7z(-z+3)-8(-z+3)[/tex]
Using distributive property, a.(b+c)=a.b+a.c
[tex](-4z^3+12z^2)+(-7z^2+21z)+(8z-24)[/tex]
Combining like terms
[tex]-4z^3+(12z^2-7z^2)+(21z+8z)-24[/tex]
[tex]-4z^3+5z^2+29z-24[/tex]
which is required product.
[tex]\text{Now compare above product with given product }-4z^3+xz^2+yz-24[/tex]
[tex]x=5, y=29[/tex]
a bowl contains 80 M&M candies If 15 are orange what fraction of the candies is NOT orange. Answer reduced to lowest terms
Evaluate: 3 x 5 3x5 when x=2 x=2 . Select one: a. 486 b. 96 c. 30 d. 7,776
Find the length of the hypotenuse of a right triangle with legs of lengths 5 and 12. If necessary, round your answer to two decimal places
The simple interest formula is I = Prt, where I represents simple interest on an amount, P, for t years at a rate of r. The equation solved for P is P = . What is the amount of money, P, that will generate $40 in interest at a 10% interest rate over 5 years? $60 $80 $90 $100
we know that
The simple interest formula is equal to
[tex]I=Prt[/tex]
where
I represents simple interest
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
in this problem we have
[tex]t=5\ years\\ P=?\\ I=\$40\\r=0.10[/tex]
[tex]I=Prt[/tex]
Solve for P
[tex]P=I/(rt)[/tex]
substitute the values
[tex]P=40/(0.10*5)=\$80[/tex]
therefore
the answer is
[tex]\$80[/tex]
Find the factorization of the polynomial below.
2x²+7x+6
A. (2x+2)(x+4)
B. (2x+2)(x+3)
C. (2x+3)(x+1)
D. (2x+3)(2x+2)
Answer:
(2x+3)(x+2)
Step-by-step explanation:
i hope this helps
Factoring help. Need step by step guidance ....
35n ^ 2 + 22n + 3
What is the vertex of y=-6(2.5-x)(x-5.5)?
Evaluate S5 for 600 + 300 + 150 + … and select the correct answer below
Answer:
Hence,
[tex]S_5=1162.5[/tex]
Step-by-step explanation:
We are asked to evaluate:
[tex]S_5[/tex]
We are given a geometric series.
( since each term of the series are in geometric progression as each term is half of the previous term of the series.
i.e. we have a common ratio of 1/2 )
Also, we know that sum of n terms in geometric progression is given by:
[tex]S_n=a(\dfrac{1-r^n}{1-r})[/tex]
where r is the common ratio.
a is the first term of the series.
Here we have:
[tex]a=600\ ,\ r=\dfrac{1}{2}[/tex]
Hence,
[tex]S_5=600(\dfrac{1-(\dfrac{1}{2})^5}{1-\dfrac{1}{2}})\\\\\\S_5=600(\dfrac{2^5-1}{2^4}})\\\\\\S_5=600(\dfrac{31}{16})\\\\\\S_5=1162.5[/tex]
The sum of the first five terms of the geometric progression 600 + 300 + 150 + … is 1162.5.
What is the ratio of geometric progression?A geometric progression is the series of numbers where the ratio of the two consecutive numbers is always the same.
As it is given to us the sequence 600, 300, 150 is a geometric progression. Therefore, the first term of the progression is 600, while the ratio between the two terms are,
[tex]r = \dfrac{300}{600} = \dfrac{1}{2}[/tex]
Now, we know that the sum of the geometric sequence of the first 5 terms can be written as where the value of the n is 5,
[tex]S=a\dfrac{(1-r^n)}{(1-r)}[/tex]
[tex]S=600\dfrac{(1-(\dfrac{1}{2})^n)}{(1-(\dfrac{1}{2}))}[/tex]
[tex]S = 1162.5[/tex]
Hence, the sum of the first five terms of the geometric progression 600 + 300 + 150 + … is 1162.5.
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Use the change of base formula to evaluate log4 20.
In how many different ways can a president vice-president and secretary be elected from a class of 15 students
what is (13x+9k)+(17x+6k) simplified
The expression (13x+9k)+(17x+6k) simplifies to 30x + 15k. This is done by adding the coefficients of the like terms which in this case are 13x with 17x, and 9k with 6k.
Explanation:The expression (13x+9k)+(17x+6k) is a mathematical expression in algebra with two variables, x and k. The goal here is to simplify the expression by combining like terms. In mathematics, 'like terms' refer to the terms that have the same variables and powers. The coefficients of the like terms can be different.
Now to simplify this expression, we add the coefficients of the like terms. The like terms here are 13x and 17x, and also 9k and 6k.
Add 13x and 17x, you get 30x. And when you add 9k and 6k, you get 15k.
So, the simplified version of the given expression (13x+9k)+(17x+6k) is 30x + 15k.
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