If you are adding two fractions that are both less than 1/2, What must be true about the sum? Give three examples to support your thinking.

we know that

1) Lily swims a [tex]100[/tex]-meter race

2) She swims the first half in [tex]27.8[/tex] seconds

3) She swims the second half in [tex]30.12[/tex] seconds

so

To find the total time to swim the whole race sum the time in the first half plus the time in second half

[tex](27.8+30.12)=57.92\ seconds[/tex]

therefore

the answer is

[tex]57.92\ seconds[/tex]

By adding the two given times, we will see that she completes the race in 57.92 seconds.

We know that:

Then the time that it took to complete the whole race is just the sum of these two times, we will get:

Total time = 27.8 seconds + 30.12 seconds = 57.92 seconds.

We can conclude that Lily swims the whole race in 57.92 seconds.

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Answer:

The bucket hold 3x the capacity of the jar

Step-by-step explanation:

First we have to write down the amount of both, the jar and the bucket, to make a small equation to see the problem mathematically.

Bucket = c liters of water

Jar = c - 8

so we can say that:

c = c - 8

Now that we have the equation, we can determine that the variable 'c' on the right is substracting to the number 8. So we can pass the 'c' to the left of the equation by adding it to the elements on the left, like this:

c + c = 8

Now, two 'c' are the same, we can add them:

2c = 8

This two on the equation of the left is multiplying the variable 'c', so we can move it to the right by dividing the number 8 with this number:

c = 8/2

If we solve this operation, we'll get:

c = 4

That means that the amount of liters of water a jar can hold is 4 liters

Jar = 4 liters

Now that we have this value, we can solve the problen. If the jar holds 8 liters of water les than a bucket, we can now represent it like this:

Bucket = Jar capacity + 8 liters

Wich can be represented like this as well:

Bucket = 4 + 8

And the result is:

Bucket = 12 liters

If we try to fit the jar's capacity on the bucket, we find that the bucket hold 3x the capacity of the jar

Let me know if you have any doubts :D

Answer:  The filled truth table for p ∨ q is shown below.

Step-by-step explanation:  We are given to fill the truth table for the p ∨ q.

We know that

If any one of p or q is true, then  p ∨ q is also true. And it is false, only when both p and q are false.

The filled truth table is as follows :

p                  q                     p ∨ q

T                  T                        T

T                  F                        T

F                  T                        T

F                  F                        F

Thus, a0 = T, a1 = T, a2 = T  and  a3 = F.

The percentage of the students writing papers will be 13%.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

The quantity of anything is stated as though it were a fraction of a hundred. A quarter of 100 can be used to express the ratio.

During study hall, 58% of the students are studying for tests, 29% of the students are doing homework, and the remaining students are writing papers.

The percentage of the students writing papers will be given by the difference of the 100% and the sum of 58% and 29%. Then we have

⇒ 100% - 58% - 29%

⇒ 100% - 87%

⇒ 13%

The percentage of the students writing papers will be 13%.

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Final answer:

The percentage of students writing papers during study hall is 13%.

Explanation:

To find out what percentage of students are writing papers during study hall, we need to find the remaining percentage after accounting for the students studying for tests and doing homework.

Since 58% of the students are studying for tests and 29% of the students are doing homework, the percentage of students writing papers can be found by subtracting the sum of these two percentages from 100%:

Percentage of students writing papers = 100% - (58% + 29%) = 13%

Therefore, 13% of the students are writing papers during study hall.

The decimal number 0.00001 can be written as a power of 10 as 1 x 10^-5. This is because the decimal is shifted five places to the right.

The student is asked to express the number 0.00001 as a power of 10. This can be achieved by understanding that the power (exponent) of 10 is equal to how many places the decimal is shifted. Here, the decimal is shifted five places to the right so we use the negative exponent -5 to denote this shift. Therefore, 0.00001 written as a power of 10 is 1 x 10^-5.

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Expressions equivalent to 'log 2 - log 6' are based on the property of logarithms that states 'log a - log b' is equivalent to 'log (a/b)'. Thus, the equivalent expression is 'log (1/3)'.

The expressions equivalent to log 2 - log 6 can be found using the property of logarithms that deals with the difference of logarithms. According to this property, log a - log b is equivalent to log (a/b). Therefore, one equivalent expression for log 2 - log 6 would be log (2/6), which can be simplified to log (1/3).

Raina received 19 phone calls on the third evening, 26 phone calls on the first evening, and 38 phone calls on the second evening.

Let's consider the number of phone calls Raina received each evening.

Let x represent the number of phone calls on the third evening.

The first evening, Raina received 7 more calls than the third evening, so the number of phone calls on the first evening is x + 7.

The second evening, Raina received 2 times as many calls as the third evening, so the number of phone calls on the second evening is 2x.

Given that Raina received a total of 83 phone calls over the three evenings, we can set up the equation:  

x + (x + 7) + 2x = 83

Combining like terms, we have:

4x + 7 = 83

Subtracting 7 from both sides, we get:

4x = 76

Dividing both sides by 4, we find that x = 19.

Therefore, Raina received 19 phone calls on the third evening, 26 phone calls on the first evening, and 38 phone calls on the second evening.

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Final answer:

By setting up a system of equations based on the given information and solving for b, we find that side a is 40 meters, b is 20 meters, and c is 38 meters.

Explanation:

To solve for the lengths of the sides of the triangle, we can set up a system of equations based on the information given:

Substituting the expressions for a and c into the perimeter equation gives:

With the value for b, we can now find a and c:

The lengths of the sides of the triangle are 20 meters (b), 40 meters (a), and 38 meters (c).

Final answer:

To solve for the lengths of the triangle's sides, we formulate equations based on the given relationships and perimeter. We find that side b is 20 meters, side a is twice b (40 meters), and side c is 2 meters shorter than a (38 meters).

Explanation:

The student's question involves finding the length of each side of a triangle given specific relationships between the sides and the overall perimeter. The problem can be formulated using algebra. Let's denote side b as the baseline. From the question, we know that side a is twice side b (a = 2b), and side c is 2 meters shorter than side a (c = a - 2 = 2b - 2). The perimeter (P) is given as 98 meters, so we have P = a + b + c = 98 meters.

Substituting for a and c in terms of b we have P = 2b + b + (2b - 2) = 98, simplifying to 5b - 2 = 98. Solving for b we add 2 to both sides, 5b = 100, and then divide by 5 to get b = 20 meters. Now we easily find that a = 2(20) = 40 meters, and c = 40 - 2 = 38 meters. Therefore, the lengths of the sides of the triangle are 40 meters for side a, 20 meters for side b, and 38 meters for side c.

Answer:

y=[tex]\frac{x^{2} }{18}[/tex]

To find the derivative of the given function using the definition of derivative, we can start by taking the limit of the expression. By simplifying the expression and canceling out terms, we can find the derivative as -10/(20x√x-√x)(10x-1/2).

The function f(x) = 1/(10x-1/2)

To find the derivative of the function using the definition of derivative, we can start by writing the expression of the derivative:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h

Substituting the function f(x) into the expression, we have:

f'(x) = lim(h->0) [1/(10(x+h)-1/2) - 1/(10x-1/2)] / h

To simplify the expression, we can find a common denominator and combine the fractions:

f'(x) = lim(h->0) [(10x-1/2) - (10(x+h)-1/2)] / h[(10(x+h)-1/2)(10x-1/2)]

Expanding and simplifying the expression, we get:

f'(x) = lim(h->0) -10/(2√x-√(x+h)) / h(10x-1/2)(10(x+h)-1/2)

Now, we can cancel out the h in the denominator:

f'(x) = lim(h->0) -10/(2√x-√(x+h))(10x-1/2)(10(x+h)-1/2)

Taking the limit as h approaches 0, we get:

f'(x) = -10/(2√x-√x)(10x-1/2)(10x-1/2)

Now, we can simplify further:

f'(x) = -10/(20x√x-√x)(10x-1/2)

So, the derivative of the function is f'(x) = -10/(20x√x-√x)(10x-1/2)

complete question given below:

Find the derivative of the function using the definition of derivative.

1 f(x) = 1/10x-1/2

f'(x) =

State the domain of the function, f(x). (Enter your answer in interval notation.)

State the domain of its derivative, f'(x). (Enter your answer in interval notation.)

Answer:

Option B is correct

Left 1 unit.

Explanation:

According to the graph theory of transformation:

y = f(x+k)=[tex]\left \{ {{k>0 shift graph of y= f(x) left k unit} \atop {k<0} shift graph of y= f(x) right |k| unit} \right.[/tex]

Given the parent function: [tex]f(x)=x^2[/tex]

and the function [tex]g(x)=x^2+2x+1[/tex]

we can write it as:

g(x)= [tex](x+1)^2[/tex]   [ ∴[tex](a+b)^2 = a^2+2ab+b^2[/tex] ]

Therefore, vertex of the graph of the function [tex]g(x)=(x+1)^2[/tex] is 1 units to the left of the vertex of the graph of the function [tex]f(x)=x^2[/tex] .

Answer:

Shift 1 unit left

B is correct

Step-by-step explanation:

Given: The vertex of f(x) shift to g(x)

[tex]f(x)=x^2[/tex]

Vertex of f(x): (0,0)

[tex]g(x)=x^2+2x+1[/tex]

Vertex form: [tex]y=a(x-h)^2+k[/tex]

[tex]g(x)=(x+1)^2[/tex]

Vertex of g(x): (-1,0)

[tex](0,0)\rightarrow (-1,0)[/tex]

Only x-coordinate change and y-coordinate remain same.

[tex]0\rightarrow -1[/tex]

Hence, The vertex of f(x) shift 1 unit left to get vertex of g(x)

The angle bisectors of the same side interior angles, formed by two parallel lines and a transversal, intersect at a 90-degree angle.

In geometry, when you have two parallel lines intersected by a transversal, the same side interior angles are those which are in the same position on the two lines. If we bisect these angles (i.e., divide them in two equal parts), the angle bisectors will intersect forming an angle that is 90 degrees.

To understand why, note that the same-side interior angles are supplementary (i.e., they sum up to 180 degrees) because they are on the same side of the transversal. So, if we bisect these angles, each pair of congruent angles will add up to 90 degrees. When these bisectors intersect, they form four right angles. Therefore, the angle at the intersection of the bisectors is 90 degrees.

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The problem primarily involves the calculation and understanding of the concepts of perimeter and area in Mathematics. Given the length is 3 ft longer than the width, and the perimeter of the rectangle is 26 feet, first, we find that the width is 5 feet and the length is 8 feet. The area is thus, 40 square feet.

To solve this problem, we need to understand the formula for the perimeter of a rectangle, which is 2l + 2w, where 'l' stands for length and 'w' for width. Also, the area of a rectangle is calculated by l*w. The problem states that the length is 3 ft longer than the width and that the perimeter is 26 feet. Firstly, we can express the length as 'w + 3' and substitute into the perimeter formula:

2(w + 3) + 2w = 26

After simplifying, we get the width equals to 5 feet. Substituting 5 for width, we get the length as 8 feet, because length is 3 feet longer than the width. Then, the area of the rectangle can be calculated by multiplying the width by the length:

Area = l * w = 8 ft * 5 ft = 40 square feet

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Answer: 825.

Step-by-step explanation: