A triangle has an area of 30 inches² and a height of 12 inches. What is the length, in inches, of the base?

Answer: To find the total change in pounds of fruit in the basket, we need to subtract the amount taken out from the amount added.

First, let's convert all the fractions to decimals:

1 1/4 = 1.25

2 3/8 = 2.375

Now, we can subtract:

1.25 - 2.375 = -1.125

So, the total change in the fruit basket was -1.125 pounds, meaning the basket had 1.125 pounds less fruit after the removal.

Step-by-step explanation:

The probability of selecting the fair coin when a gambler has both a fair coin and a two-headed coin in his pocket is 50%.

This is because, when he selects one of the coins at random, it is equally likely that he has chosen either the fair coin or the two-headed coin. If he flips the coin and it shows heads, then the probability that it is the fair coin is still 50%, because either coin could have been selected and both would show heads.This is an example of a classical probability problem, which is based on the law of equal chances. This law states that when an event has multiple possible outcomes, each of those outcomes is equally likely to occur. In this case, there are two possible coins that could have been chosen, so each coin has a 50% chance of being selected. This means that the probability of selecting the fair coin is 50%.This also applies to events that have multiple outcomes which are not equally likely. For example, if a gambler has a fair coin, a two-headed coin, and a three-headed coin in his pocket, the probability of selecting the fair coin is still 50%. This is because the gambler has an equal chance of selecting any of the coins, regardless.

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Answer:

D. 2x^3−6x^2−x

Step-by-step explanation:

Factor the numerator and denominator and cancel the common factors.

Hope this helps.

The first day after the autumn equinox that the solar panel generates 8400 kJ is approximately 39.2 days, or 0.107 radians, after the equinox.

To find the first day after the autumn equinox that the solar panel generates 8400 kJ, we need to solve the equation:

624sin(2π/365)t + 8736 = 8400

Simplifying the equation, we get:

sin(2π/365)t = 0.958

Taking the inverse sine of both sides, we get:

(2π/365)t = 1.296 or (2π/365)t = 3.846

Solving for t in each equation, we get:

t ≈ 78.97 or t ≈ 235.62

Since t is measured in radians, we need to convert it to days by dividing by 2π/365:

t ≈ 78.97 / (2π/365) ≈ 86.75 or t ≈ 235.62 / (2π/365) ≈ 256.73

Therefore, the first day after the autumn equinox that the solar panel generates 8400 kJ is either day 87 or day 257.

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Answer:

∴Sum of interior angles in ΔABC = 180°

∠A + ∠B + ∠C = 180°

2x° + 4x° + 6x° = 180°

              12x° = 180°

                = x =

                ∴x = 15

∴∠C = (6)×(15)

      = 90°

This means AC makes a 90° angle (i.e. ∠C) with BC.

∴AC is perpendicular to BC (Proved)

Also ΔABC is a right-angled triangle

The value of a in the equation a + 2 = 3 is 1.

An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.

It is important to note that an equation is the mathematical statement which can be made up of two expressions which are connected by an equal sign.

a + 2 = 3

Subtract 2 from both sides

a + 2 - 2 = 3 - 2

a = 1

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Answer:

5 < x <  15

Step-by-step explanation:

The triangle inequality theorem states that the sum of the measures of any two sides of a triangle must be greater than the measure of the third side

In the given triangle we are provided measures of two of the sides as 10 and 5

Let the measure of the third side be x

So the three sides are 10, 5 and x

Then by the inequality theorem
10 + 5 > x

==>  15 > x or

x < 15    This is an upper bound for x

when we switch sides in an inequality > changes to < and < changes to >

We also have
x  + 5 > 10  ==> x > 10 - 5 ==> x > 5

and

x + 10 > 5 ==> x > -5

Since x > 5 is more restrictive than x > -5, we conclude that x > 5 or 5 < x is the lower bound on x

Combining all inequalities we get

5 < x <  15

Note

We could also state the lower and upper bound limits as

difference of two sides < x < sum of two sides
10 - 5 < x < 10 + 5

or

5 < x < 15

While this may seem easier to compute than the explanation given above, the derivation is left out and may confuse some students

The surface area of a cylinder with radius 5.9 ft and height 4.4 ft is  381.8 square feet.

a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

We have to find the  the surface area of a cylinder with radius 5.9 ft and height 4.4 ft.

The surface area of cylinder=2πr(r+h)

r=5.9

h=4.4

The value of pi is 3.14

The surface area of cylinder=2×3.14×5.9(5.9+4.4)

=37.052(10.3)

=381.8 square feet.

Hence, the surface area of a cylinder with radius 5.9 ft and height 4.4 ft is  381.8 square feet.

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Answer:

R1008

Step-by-step explanation:

720 X 1.4 = R1008

I used 1.4 because there is a 40 percent profit

The instantaneous rate of change of R(t) at time t is the limit of the average rate of change of R(t) as h approaches 0. In this case, the instantaneous rate of change of R(t) at time t = 1 is R'(1) = 9000 dollars/day.

The average rate of change of a function is the ratio of how much the output of the function changes over a given period to the length of that period. We can calculate this for the function R(t) = 210 + 30t3 by considering the time intervals [t, t+h] at h = 1, 0.1 and 0.01 days.

For h = 1 day, the average rate of change of R(t) is 90 dollars/day. This can be calculated by dividing the difference in output of the function at t and t+h, i.e., 30t3+30(t+1)3 = 1080, by the length of the period, h = 1 day.

For h = 0.1 day, the average rate of change of R(t) is 900 dollars/day. This is calculated by dividing the difference in output of the function at t and t+h, i.e., 30t3+30(t+0.1)3 = 9, by the length of the period, h = 0.1 day.

For h = 0.01 day, the average rate of change of R(t) is 9000 dollars/day. This is calculated by dividing the difference in output of the function at t and t+h, i.e., 30t3+30(t+0.01)3 = 0.9, by the length of the period, h = 0.01 day.

The instantaneous rate of change of R(t) at time t is the limit of the average rate of change of R(t) as h approaches 0. In this case, the instantaneous rate of change of R(t) at time t = 1 is R'(1) = 9000 dollars/day.

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Note that the lenght of segment LN = 55.14 (Option D)

This is arrived at using the Chord Bisector Theorem and the  lengths of the sides of a right-angled triangle.

According to the theorem, the line that goes through the circle's center and is perpendicular to the chord also bisects it.

This means that LK (as per the attached image) is the same as MK.

The Right Angle Postulate also means that:

LX = √[KX² - LK²]

LX = √[29² - 9²]

LX = √(841  81)

LX = √760

LX = 27.57

Since LN = LX *2
LN = 27.57 * 2

LN = 55.14

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Answer: 12

Step-by-step explanation:

we divide 156 by 13, however, we can stop as soon as we reach the tenths column as we only care about full glasses.

if done correctly, you get 13

The required Venn diagram is given in the picture.

The graphic representation of the differences and affinity between the two concepts is a Venn diagram.

In order to solve problems based on these sets, we can use a Venn diagram to depict the logical relationship among sets and their contents.

Note: "The set of natural numbers starts from 1, hence '0' is not included in the set of natural numbers."

Here we have

A spinner is split into eight wedges numbered 0-7

The set of numbers = 0, 1, 2, 3, 4, 5, 6 and 7  

The set of numbers that is less than 5 = {0, 1, 2, 3, 4 }

The set of numbers that are natural numbers = {0, 1, 2, 3, 4, 5, 6, 7 }

The set number that natural number and less than 5 = {1, 2, 3, 4 }  

Hence,

The required Venn diagram is given in the picture.

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[ Complet Question is given in picture ]

Amount of iced-tea consumed during party was 92 cups, which is equivalent to 23 quarts. It's important to note difference between volume units such as cups and quarts, and the conversion factor between them.

To determine the number of cups of iced-tea that were consumed during the party, we need to subtract the amount that remained from the total amount that was in the drink dispenser at the start of the party.

One cup of iced tea is equal to 1/16 of a quart, so 7 cups is equal to

7 * (1/16) = 7/16 quarts.

Thus, the total amount of iced tea that was consumed during the party is 912 quarts -

7/16 quarts = 912 - 7/16

= 912 - 7 * (1/16)

= (912 * 16 - 7 * 1) / 16

= 1476 / 16

= 92.25 cups, rounded down to the nearest whole number, which is 92 cups.

It is important to note that the answer is given in cups and not quarts. Quarts is a unit of volume and cups is a unit of volume as well, but they are not equivalent. 1 quart is equal to 4 cups. If we wanted to express the amount of iced tea consumed in quarts, we would multiply the answer by 1/4.

In conclusion, the number of cups of iced tea consumed during the party is 92 cups, and if we express this in quarts it is equivalent to

92 * (1/4) = 23 quarts.

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Answer:

y = -17/18x + 5/18

Step-by-step explanation:

isolate the y so add five to the other side
18y = 5 + -17x

divide everything by 18

18y/18 -17x/18 5/18
y= -17/18x + 5/18

The missing side length is x = 30

What are basic Trigonometric functions?

The angles of sine,  cosine, and tangent are the primary classification of functions of trigonometry. And the three functions which are cotangent, secant, and cosecant can be derived from the primary functions. Basically, the other three functions are often used as compared to the primary trigonometric functions.

In a Right-angled triangle, the tan of an angle is equal to the opposite side/adjacent side

from the given figure,

tan 37° = x/40

0.75 = x/40

x = 30

Therefore, The missing side is 30.

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The system of equations has infinite solutions.

Given:

y = -6x +2

-12x - 2y= -4

To solve for Equation 2,

The value of y is already given in equation 1,

Thus,

substituting the value of y in equation 2,

-12x -2(-6x +2) = -4

-12x - 12x = -4 +4

0=0

The solution of the two equations is 0. Also, we can see that both equations are in ratio.

Further, the image also shows that the line of the two equations are coinciding.

Hence, the system of equations has infinite solutions.

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How many solutions does this linear system have?

y = -6x +2

-12x - 2y= -4

one solution: (0, 0) one solution: (1, –4) no solution infinite number of solutions.

Answer: 12

she has 15 cups of sugar and needs 1.25 cup for a batch so we say 15/1.25 to get 12

Answer:

12

Step-by-step explanation:

15 / 1.25 = 12 she can make 12 batches

Jonathan's acceleration is -0.3043 m/s² East, which means he is slowing down.

Acceleration is the rate of change of the velocity of an object with respect to time.

Jonathan's initial velocity, u = 12 m/s East

His final velocity, v = 5 m/s East

The time taken, t = 23 seconds

We can use the formula for acceleration:

a = (v - u) / t

Substituting the given values, we get:

a = (5 m/s - 12 m/s) / 23 s

a = -7 m/s / 23 s

a = -0.3043 m/s^2 East

Therefore, Jonathan's acceleration is -0.3043 m/s² East, which means he is slowing down.

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Step-by-step explanation:

This is a family of curves represented by the equation:

y = a (x + 4) (x − 2)

Different values of the a coefficient result in different quadratic functions.

The vertex of the parabolas is halfway between the zeros, at x = -1.

y = a (-1 + 4) (-1 − 2)

y = -9a

So to find the value of a, divide the y-coordinate of the vertex by -9. For example, curve A has a vertex at (-1, -3), so the value of the a coefficient is -3 / -9 = ⅓.

A. y = ⅓ (x + 4) (x − 2)

B. y = (x + 4) (x − 2)

C. y = 2 (x + 4) (x − 2)

D. y = -⅓ (x + 4) (x − 2)

E. y = -2 (x + 4) (x − 2)

The solution of the system is the set of all points (x, y) that satisfy both equations, and the points in the solution set are (-1, 3) and (2, -2).

What is System of Equation?

A system of equations is a collection of one or more equations with several variables. The variable mappings that satisfy all component equations—that is, the places where all of these equations intersect—are the solutions of systems of equations.

To solve the system of equations graphically, we can plot each equation on the same set of axes.

Here is how to solve the system of equations:

y = -x² + 4x - 2

4x + 2y = 12

The points of intersection between the two lines are the solutions of the system of equations. We can see from the graph that there are two points of intersection, at approximately (-1, 3) and (2, -2).

Therefore, the solution of the system is the set of all points (x, y) that satisfy both equations, and the points in the solution set are (-1, 3) and (2, -2).

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The equation that represents the situation is q = 40 - 7

The formula for the perimeter of a quadrilateral is expressed as;

P = a + b + c + d

Where the sides, a , b , c and d are all the sides of the quadrilateral

From the information given, we have that;

The sides are a, the fourth is 7

The perimeter = 40 inches

Substitute the values

40 = 7 + q

collect the like terms, we get

q = 40 - 7

subtract the values

q = 33 inches

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In geometry, two triangles are said to be congruent if they are the same size and shape. This means that all corresponding angles and sides are equal.

To prove that two triangles are congruent, we must use one of the five congruency tests: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL).The SSS test states that if all three sides of two triangles are equal, then the triangles are congruent. The SAS test states that if two sides and the included angle of two triangles are equal, then the triangles are congruent. The ASA test states that if two angles and the included side of two triangles are equal, then the triangles are congruent. The AAS test states that if two angles and a non-included side of two triangles are equal, then the triangles are congruent. Lastly, the HL test states that if the hypotenuse and a leg of a right triangle are equal to the hypotenuse and a leg of another right triangle, then the triangles are congruent.In order to prove that two triangles are congruent, we must show that one of the five congruency tests is true. To do this, we can use techniques such as the Reflexive Property, the Symmetric Property, and the Transitive Property. We can also use logical reasoning, such as if two angles are equal and their corresponding sides are proportional, then the triangles must be congruent.

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What is the definition of congruent triangles?

By applying standard deviation concept, it can be concluded that all the students had the same score.

The mean is the average of all the scores. It is calculated by adding up all the scores and dividing them by the number of scores.

The standard deviation is a measure of how spread out the scores are. It is calculated by finding the difference between each score and the mean, squaring those differences, finding the average of those squared differences, and then taking the square root of that average.

If all the scores are the same, then the difference between each score and the mean will be 0. This means that the standard deviation will also be 0. Therefore, the correct answer is "all the students had the same score."

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In the triangle QRS the sum of square of 6 and square root of 85 is equal to 11 and RT is height.

What is triangle ?

Triangle can be defined in which it consists of three sides , three angles and sum of three angles is always 180 degrees.

Given ,

AQRS is an isosceles triangle. Complete the statement that explains whether RT is the height of the triangle.

So, from the given triangle we can say that ,

[tex]6^{2}[/tex] + [tex]\sqrt{85 }^2[/tex] = [tex]11^{2}[/tex]

By the pythagorean theorem , we can say that ,

the sum of squares of base and opposite side is always equals to square of hypothenuse.

so it is equal to square of 11 and RT is height.

Therefore, in the triangle QRS the sum of square of 6 and square root of 85 is equal to 11 and RT is height.

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The lines 4x+y=5 and 3x+12y=-6are neither parallel nor perpendicular

From the question, we have the following parameters that can be used in our computation:

4x+y=5 3x+12y=-6

Make y the subject in the equations

So, we have

y = -4x + 5

y = -1/4x - 1/2

The slopes of the lines are -4 and -1/4

These numbers are not equal and they are not opposite reciprocals

This means that the lines are neither parallel nor perpendicular

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The algebraic expression in simplified form representing the average is 3x + 1.

To solve this question, one must know the formula of average. It is as follows -

Average = sum of the numbers / quantity of numbers

Now, we see that there are 3 expressions in the question. Hence, quantity of numbers will be 3.

Sum of the numbers = 2x + 3 + x + 6x

So, keep the values in formula to find the expression -

Average = 2x + 3 + x + 6x/ 3

Simplifying the expression to find average

Average = 9x + 3/3

Performing division on Right Hand Side of the equation

Average = 3x + 1.

Hence, required algebraic expression is 3x + 1.

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It should be noted that the set of odd integers has the density property. The density property states that for every subset of the real numbers that contains an infinite number of points, the interior of that subset contains a non-empty open interval. This is true

The set of odd integers is a countable subset of the real numbers, meaning that it can be put into a one-to-one correspondence with the natural numbers. However, even though it is countable, it still has an infinite number of points.

Since the set of odd integers contains an infinite number of points, it must contain a non-empty open interval in its interior. To see this, consider the interval (2n-1, 2n+1) for any positive integer n. This interval contains the odd integer 2n and its endpoints are both odd integers, so it is completely contained in the set of odd integers.

Therefore, the set of odd integers has the density property and is a dense subset of the real numbers.

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The depth of an Olympic pool that goes to 50 ft. deep can be expressed as:

d = 50 ft.

Olympic pool is a type of swimming pool that is used for competitive swimming and diving events. These pools typically have a uniform depth, a smooth and flat bottom and multiple lanes for swimmers to compete in.

The depth of an Olympic pool that goes to 50 ft. deep can be expressed as:

d = 50 ft.

Where d is the depth of the pool. This expression simply states that the depth of the pool is equal to 50 ft.

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Yes!

[tex](2^{2} )^{3}[/tex] = 2^6 = 64

(8-4)^2 = 4^2 = 16

then for the last part

4^-4 is the same as [tex]1/4^{4}[/tex] or 1/256

Then it is order of operations:

do 16 x 1/256 first which is 1/16

and then 64 - 1/16 to get 63 15/16 :)