The given limit problem involves factorization of difference of squares. After simplifying, the limit turns out to not exist as the denominator becomes zero while the numerator becomes non-zero. Hence, we end up with an undefined expression.
Explanation:The subject of this question is the calculation of a mathematical limit, a fundamental concept in calculus. The limit equation given is a case of an indeterminate form of type 0/0, which can be solved using the L'Hopital's Rule.
However, before that, we should identify the factorization of (x4 - y4) and (x2 - y2) which are a difference of squares. Therefore, they can be factored as follow: (x4 - y4) = (x2 + y2)(x2 - y2) and (x2 - y2) = (x - y)(x + y).
After canceling out the common factor (x2 - y2), we get the simplified expression (x2 + y2)/(x - y). Now, considering that x≠y, we can substitute x and y with 'a' (since both approach the same value 'a' as per the limit definition), and we obtain (2a)/(0) = ∞ or -∞ based on a's sign. So, in fact, this limit does not exist for x≠y.
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Will the standard form 3.2 × 10–4 be more or less than 1? Explain what effect the negative exponent has. Thanks! =)
Nail Polish. A Specific shade of red nail polish requires 7 parts red to 2 parts yellow. A mixture contains 45 quarts. How many quarts of red? How many quarts of yellow?
By setting up a proportion using the given ratio of 7 parts red to 2 parts yellow, we can determine that in a 45-quart mixture, there will be 35 quarts red and 10 quarts yellow.
Explanation:To find out the number of quarts for each color in the mixture, we need to set up a proportion. The given ratio of red to yellow is 7:2. So, for every 9 parts (7 red + 2 yellow), we have a specified amount of nail polish.
The total amount of nail polish we have is 45 quarts. Each 'part' of our ratio can then be calculated as 45 (total quarts) divided by 9 (total parts), which equals 5 quarts per part.
So, for the red nail polish, since it's 7 parts, we have 7 parts * 5 quarts/part = 35 quarts. And for the yellow nail polish, as it's 2 parts, we have 2 parts * 5 quarts/part = 10 quarts.
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Approximately 105 out of 350 teens have their own savings account. Express the ratio 105:350 as a fraction in simplest form.
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Find the solution of this system of equations
-3x-7y=-66
-10x-7y=-24
Dominic has $15 for dinner. His meal costs $13.90. He wants to leave an 18% tip. Does he have enough money? Explain your reasoning
Triangle PQR has sides measuring 9 feet and 10 feet and a perimeter of 24 feet. What is the area of triangle PQR? Round to the nearest square foot.
square feet
Answer:
22 square foot
Step-by-step explanation:
Consider PQR be a triangle, such that PQ=9 feet, PR=10 feet.
Now, Perimeter of triangle=24
⇒Sum of all the sides=24
⇒PQ+PR+QR=24
⇒9+10+QR=24
⇒QR=5 feet
Also, s=[tex]\frac{a+b+c}{2}=\frac{9+10+5}{2}=12[/tex]
Area of triangle A=[tex]\sqrt{s(s-a)(s-b)(s-c)}[/tex]
=[tex]\sqrt{12(12-9)(12-10)(12-5)}[/tex]
=[tex]\sqrt{12(3)(2)(7)}[/tex]
=[tex]\sqrt{504}[/tex]
=[tex]22.44[/tex]
≈[tex]22sq foot[/tex]
thus, area of triangle=22 square foot.
A 21-inch piece of steel is cut into three pieces so that the second piece is twice as long as the first piece, and the third piece is three inches more than six times the length of the first piece. Find the lengths of the pieces
Eben rolls two standard number cubes 36 times. Predict how many times he will roll a sum of 4.
Charlie, who is 4 feet tall, walks away from a streetlight that is 15 feet high at a rate of 6 feet per second, as shown in the figure. Express the length s of Charlie's shadow as a function of time. (Hint: First use similar triangles to express s as a function of the distance d from the streetlight to Charlie.)
which equation best represents the model?
Maria incorrectly placed the decimal point when she wrote 0.65 inch fo the width of her computer. what is the correct decimal number for the width?
Answer:
It should be placed after 6 it means 6.5inch
Step-by-step explanation:
Maria should placed the decimal point after 6 it means 6.5 inch. Because the order of the width of her computer should be in between 6 inch to 9 inches.
Width of a computer cannot be 0.65 inches because it will be too short and we cannot called it computer on the other hand if she put the decimal after 5 then the width of her computer will be 65 inches which is not normal.
(–0.2b^6)^3(5b) this is the question to be answered
Final answer:
The simplified form of the expression [tex](-0.2b^{6})^3(5b)[/tex] is [tex]-0.04b^{19}[/tex], obtained through raising to a power and multiplication.
Explanation:
The student's question involves simplifying an algebraic expression by raising a term to a power and then multiplying it by another term. To simplify [tex](-0.2b^{6})^3(5b)[/tex], we must first raise [tex]-0.2b^{6}[/tex] to the third power and then multiply the result by 5b. This will give the answer [tex]-0.04b^{19}[/tex], which can be obtained following the below mentioned steps.
Let's do this step by step:
Raise [tex]-0.2b^{6}[/tex] to the third power: [tex](-0.2)^3[/tex]= −0.008 and [tex](b^{6})^3[/tex]= [tex]b^{18}[/tex]
Multiply the result by 5b: [tex](-0.008)b^{18}*5b[/tex] = [tex](-0.008*5)b^{18+1}[/tex] = [tex]-0.04b^{19}[/tex]
The final simplified expression is [tex]-0.04b^{19}[/tex].
Consider that lines u and v are parallel. Which equation models the relationship between the angles? What is the value of x? A) 12x - 4 = 10x + 10; x = 7 B) 12x - 4 + 10x + 10 = 180; x = 7.9 C) 12x - 4 = 10x - 10; x = -3 D) 12x + 4 = 10x + 10; x = 3
Answer:
A) 12x - 4 = 10x + 10; x = 7
Step-by-step explanation:
12x - 4 = 10x + 10; x = 7
Since lines u and v are parallel, the two angles are corresponding angles. Corresponding angles are equal to each other.
12x - 4 = 10x + 10
Solve for x:
2x = 14
x = 7
Richard borrowed $1,250 for two years at 14% a year under an add-on plan. He repaid the loan, including interest, in 24 equal payments. How much was each payment?
Let f(x)=12x2+4. The function g(x) is a vertical stretch of f(x) by a factor of 4. What is the equation of g(x)?
Answer:
The new function is [tex]g(x)=48x^2+16[/tex]
Step-by-step explanation:
We are given,
The function [tex]f(x)=12x^2+4[/tex] is vertically stretched by a factor of 4.
Now, we know that,
Vertical stretch changes the function f(x) to [tex]kf(x)[/tex] where k is the scale factor.
So, we get,
The new function is [tex]g(x)=4f(x)[/tex]
i.e. [tex]g(x)=4(12x^2+4)[/tex]
i.e. [tex]g(x)=48x^2+16[/tex]
Thus, the equation of the new function is [tex]g(x)=48x^2+16[/tex].
f the centripetal and thus frictional force between the tires and the roadbed of a car moving in a circular path were reduced, what would happen?
Jamie had 5 1/4 cans if soup in her kitchen. She decided to use 3 1/9 of it. How much soup does she still have after making 3 1/9 if it?
please explain
Two arrows are launched at the same time with the same speed. arrow a at an angle greater than 45 degrees, and arrow b at an angle less than 45 degrees. both land at the same spot on the ground. which arrow arrives first?
Verify stokes' theorem for the helicoid ψ(r,θ)=⟨rcosθ,rsinθ,θ⟩ where (r,θ) lies in the rectangle [0,1]×[0,π/2], and f is the vector field f=⟨6z,8x,8y⟩. first, compute the surface integral: ∬m(∇×f)⋅ds=∫ba∫dcf(r,θ)drdθ, where a= , b= , c= , d= , and f(r,θ)= (use "t" for theta). finally, the value of the surface integral is . next compute the line integral on that part of the boundary from (1,0,0) to (0,1,π/2). ∫cf⋅dr=∫bag(θ)dθ, where a= , b= , and g(θ)=
Stokes' theorem, which relates a surface integral of a curl of a vector field over a surface to a line integral of the vector field over the boundary of the surface, can be applied to verify a surface described by a helicoid and a given vector field, through calculation of the surface and line integrals, even when specific function values are not provided.
Explanation:Stokes' theorem relates a surface integral of a curl of a vector field over a surface Ψ to a line integral of the vector field over the boundary ∂Ψ of the surface. Given a helicoid ψ(r,θ)=⟨rcosθ,rsinθ,θ⟩ where (r,θ) lie in the rectangle [0,1]×[0,π/2], and f is the vector field f=⟨6z,8x,8y⟩, Stokes' theorem can be applied to verify the vectro field over the given area.
The process involves two primary steps: computation of the surface integral, and computation of the line integral.
Step 1: Compute the surface integral ∬m(∇×f)⋅ds=∫ba∫dcf(r,θ)drdθ. However, because the specific values for a, b, c and d, and the function f(r,θ) are not defined in this question, the exact calculation can't be provided.
Step 2: Compute the line integral ∫cf⋅dr=∫bag(θ)dθ, on the boundary from (1,0,0) to (0,1,π/2). Again, specific values for a, b and the function g(θ) are not provided.
According to Stokes' theorem, the two results should be equal.
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Greta completed a mile race in 5 minutes . inez ran a mile in which each quarter mile split was 1 min 20 seconds which of the two girls had the fastest time? How much faster?
The graph shows a line and two similar triangles.
Answer:
Option (a) is correct.
The equation of the line is expressed using expression [tex]\frac{y}{x}=\frac{1}{4}[/tex]
Step-by-step explanation:
Given : The graph shows a line and two similar triangles.
We have to find the expression that finds the equation of line.
Since, given two triangles are similar.
So, Δ ABC ≅ Δ ADE
Thus, There corresponding sides are in same proportion.
[tex]\frac{AC}{AD}=\frac{CB}{DE}= \frac{AB}{BE}[/tex]
Substitute, we get,
[tex]\frac{1}{y}=\frac{4}{x}= \frac{AB}{BE}[/tex]
Rearrange, we have,
[tex]\frac{1}{4}=\frac{y}{x}= \frac{AB}{BE}[/tex]
Also, finding slope of line AE,
Coordinate of B is (4,1) and Coordinate of A is (0,0)
Th equation of line is y = mx + c
Where, m is slope and c is x intercept
Since, [tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{1-0}{4-0}[/tex]
Simplify, we have,
[tex]m=\frac{1}{4}[/tex]
And c = 0
Thus, equation of line is[tex]y=\frac{1}{4}x[/tex]
We re-writing we get,
[tex]\frac{y}{x}=\frac{1}{4}[/tex]
Thus, The equation of the line is expressed using expression [tex]\frac{y}{x}=\frac{1}{4}[/tex]
Two trucks were driven on a 1,680 kilometer (km) trip. the first truck averaged 14 km per liter of fuel for the trip, and the second averaged 12 km per liter. the second truck used how many more liters of gas than the first?
The second truck used 20 liters more gas than the first truck.
It is given that two trucks were driven. First truck averages 14 km per liter of fuel whereas the 2nd truck averages 12 km per liter of fuel.
We have to find that how many more liters of gas the 2nd truck used in comparison to 1st truck.
What will be the value if we subtract 20 from 40 ?
The value will be 20.
Fuel used by the first truck will be ;
1680 / 14 = 120 liters
Fuel used by second truck will be ;
1680 / 12 = 140 liters
Difference of fuel used by 1st truck and 2nd truck is
140 - 120 = 20 liters
Thus the second truck used 20 liters more fuel than the first.
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(h) when is the particle speeding up? (enter your answer using interval notation.) (2,4)∪(6,8) incorrect: your answer is incorrect. f(t) = cos(πt/4)
sosceles triangle LMN is graphed with vertices L(0, 1), M(3, 5), and N(6,1). What is the slope of side LN?
Answer:
0
Step-by-step explanation:
the other person is correct! :D
The Leukemia and Lymphoma Society sponsors a 5K race to raise money. It receives $55 per race entry and $10,000 in donations, but it must spend $15 per race entry to cover the cost of the race. Write and solve an inequality to determine the number of race entries the charity needs to raise at least $55,000.
Find the constant of variation k for the direct variation
A) k=2.5
B) k=-2
C) k=2
D) k=0.5
example for empirical probability
Empirical probability is a form of probability that is based on the actual results of an experiment. It's computed by dividing the number of times an event occurs by the total number of observations or trials. Therefore, empirical probability varies depending on the outcomes of the experiment.
Explanation:The empirical probability, or experimental probability, comes from actual observations or experiments, unlike theoretical probability which is based purely on mathematical principles. An example of empirical probability can be found in a simple coin toss experiment. Let's say we toss a coin 100 times and heads comes up 55 times.
To calculate the empirical probability of getting heads, we would divide the number of times the event (getting heads) occurs by the total number of opportunities for the event to occur (the total number of tosses). In this case, the empirical probability is given by 55 (the occurrences of heads) divided by 100 (total coin tosses), giving us an empirical probability of 0.55 for getting heads.
Another example, in a traffic situation, would be to install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a certain period of time. This data would allow us to calculate the empirical probability of a car running the red light.
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The base and height of Triangle A are half the base and the height of Triangle B. How many times greater is the area of Triangle B?
um plz help On a soccer team, 11 out of 17 players surveyed say they had two or more siblings. The league has 850 players. Which is the best prediction of the number of players in the league that have two or more siblings?
Match the equation with the step needed to solve it.
1 .
subtract m
2m - 1 = 3m
2 .
add 2
2m = 1 + m
3 .
subtract 2m
m - 1 = 2
4 .
add 1
2 + m = 3
5 .
subtract 2
-2 + m = 1
6 .
subtract 1
3 = 1 + m
Answer:
1.
add 1
m - 1 = 2
2.
subtract 1
3 = 1 + m
3.
add 2
-2 + m = 1
4 .
subtract m
2m = 1 + m
5 .
subtract 2
2 + m = 3
6 .
subtract 2m
2m - 1 = 3m