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General Idea:

Arithmetic sequence is a sequence which have common difference. The [tex] n^{th} [/tex] term of arithmetic sequence is given by the below formula:

[tex] a_n=a_1+(n-1)\cdot d\\ \\ Where:\\ a_1 \; is \; the \; first\; term\; of\; the\; sequence\\ n is \; the\; number \; of\; terms\; in\; the\; sequence\\ d\; is\; the\; common\; difference\; =\; a_2-a_1 [/tex]

Applying the concept:

47,42,37,32, …

[tex] a_1=47\\ \\ a_2=42\\ \\ d=a_2-a_1=42-47=-5\\ \\ a_n=47+(n-1) \cdot -5\\ a_n=47-5(n-1)\\ Distributing \; 5\; inside \; the \; parenthesis\\ \\ a_n=47-5n+5\\ Combine\; like\; terms\\ \\ a_n=52-5n [/tex]

Conclusion:

The explicit rule for the sequence 47,42,37,32 is [tex] a_n=52-5n [/tex]

When using a Venn diagram to determine how many people like either funnel cake or fried Oreos out of a survey of 50, after subtracting the 20-person overlap from each individual preference to avoid double counting, the total number liking either is found to be 35 people.

To determine how many people like either funnel cake or fried Oreos, we need to use the concept of set union from Venn diagrams. First, we draw two overlapping circles, one for people who like funnel cake and another for those who like fried Oreos.

The overlapping area represents people who like both. Given that 30 people like funnel cake and 25 like fried Oreos, with a 20-person overlap liking both, we add the individual preferences and subtract the overlap to avoid double counting.

The representation is:

Number liking funnel cake only: 30 - 20 = 10

Number liking fried Oreos only: 25 - 20 = 5

Number liking both: 20

Adding these, the total number liking either funnel cake or fried Oreos is 10 + 5 + 20 = 35 people.

The graph for these inequalities intersects at point (0,0).

The graph displays the solution to the system of inequalities: 2x + y < 0 and y ≥ -4 - 2x. Here's a breakdown:

First Inequality (2x + y < 0):

This translates to a blue line with a negative slope. Points like (0, -2) and (3, -6) lie on this line.
We shade the region below this line because the inequality specifies "less than." Points there, like (1, -5), satisfy the inequality.
Second Inequality (y ≥ -4 - 2x):

This translates to a green line with a negative slope as well. Points like (0, -4) and (2, -8) lie on the line.
Unlike the first, we shade the region above this line because the inequality specifies "greater than or equal to." Points like (1, -3), above the line, satisfy the inequality.
Combined Solution:

The solution space is the overlap where both inequalities hold true. This is the triangular region shaded in both blue and green. Points within this region, like (1, -5), satisfy both inequalities.
Checking Points:

If unsure about shading, test points outside the shaded region. For example, point (0, 0) lies above both lines, satisfying both inequalities. This confirms the shading is correct.

No!  Triangle ABC is not similar to triangle ADE.

Triangle ABC is not similar to triangle ADE because the sides of triangle ADE is not corresponding to that of Triangle ABC.

Similar triangles are triangles that share the same shape but may differ in size. This means that while their corresponding angles are equal, their corresponding sides are proportional.

For two triangles to be considered similar:

In practical terms, this means that if you were to resize one triangle while maintaining the same angle measurements, you could transform it into the other triangle. The concept of similarity plays a crucial role in various mathematical fields, including geometry, trigonometry, and calculus. It enables us to solve problems involving indirect measurements, scale drawings, and geometric relationships.

Answer:

Reducing the length of the loan and increasing the down payment

Step-by-step explanation:

Assuming Francis has the ability to choose the length of his loan and the amount of the down payment.

The method that would help Francis to decrease the total cost of the loan is - reducing the length of the loan and increasing the down payment.

When a person pays more amount in down payment, then he will have to loan out a smaller amount. And he will have to pay less interest. If the loan is short, the interest to be paid will be less. If the loan term is long, the interest also increases. One has to pay interest for the whole tenure of the loan.

So, if someone wants to reduce the loan amount, he must pay more in down payment.

No, the classmate is not correct. The difference between (x+2)² and (x²+4) is 4x.

No, your classmate is not correct. The difference between (x+2)² and (x²+4) is not necessarily 0. To find the difference, we need to expand both expressions and then simplify:

(x+2)² = x² + 4x + 4

(x²+4) = x² + 4

So, the difference between the two expressions is (x² + 4x + 4) - (x² + 4) = 4x. Hence, the difference is not 0.

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Answer:

a) 30 pages b) 22.5 or 45/2 c)50/3 or 16.67 d) 1/8 e) 73/3 or 24.33 f)1/9 g) 1/27

Step-by-step explanation:

a) If Natalie has read 3/4 then all that's left is 1/4 of the book to read a whole book since 3/4+1/4=4/4=1

1/4 of 120=30 pages

b) One half of 45 is the same as dividing it by two or multiplying by  1/2.

1/2*45=45/2 =22.5

c) One third of 50, is the same of multplying by 1/3 then 1/3*50≈16.67

d) Take a look back at letter b. One half (1/2) of 1/4 is :

[tex]\frac{1}{2}*\frac{1}{4}=\frac{1}{8}[/tex]

e) One third of something, is 1/3 times something like previous cases above so 1/3*73=73/3≈24.33

f) One third of One third, is 1/3 times 1/3:

[tex]\frac{1}{3}*\frac{1}{3}=\frac{1}{9}[/tex]

g) One third of 1/9 is 1/3 times 1/9:

[tex]\frac{1}{3}*\frac{1}{9}=\frac{1}{27}[/tex]

Do not forget the rules of multiplying fractions. Numerator times Numerator, Denominator times Denominator. When we read a third, half,.. of this means one half times

Answer:  The required measures of two angles one positive and negative that are co-terminal with [tex]\dfrac{\pi}{2}[/tex] are [tex]-\dfrac{3\pi}{2}[/tex] and [tex]\dfrac{5\pi}{2}.[/tex]

Step-by-step explanation:  We are given to find measures of two angles one positive and negative that are co-terminal with [tex]\dfrac{\pi}{2}.[/tex]

Let x and y represents the positive and negative co-terminal angles respectively.

We know that

the measures of any two co-terminal angles differ from each other by a multiple of [tex]2\pi.[/tex]

So, the values of x and y can be found as follows :

[tex]x=\dfrac{\pi}{2}-2\pi=-\dfrac{3\pi}{4},\\\\\\y=\dfrac{\pi}{2}+2\pi=\dfrac{5\pi}{2}.[/tex]

Thus, the required measures of two angles one positive and negative that are co-terminal with [tex]\dfrac{\pi}{2}[/tex] are [tex]-\dfrac{3\pi}{2}[/tex] and [tex]\dfrac{5\pi}{2}.[/tex]

Final answer:

The positive coterminal angle with π/2 is 5π/2, and the negative coterminal angle is -3π/2; both angles are found by adding and subtracting 2π radians from the original angle.

Explanation:

To find the measures of two angles, one positive and one negative, that are coterminal with π/2, we need to add and subtract 2π (360 degrees) to the original angle since the angles are coterminal when they share the same initial and terminal sides and differ by a full circle (360 degrees) or a multiple thereof. The angle π/2 radians is equivalent to 90 degrees, so adding 2π radians (360 degrees) gives us a positive coterminal angle, and subtracting 2π radians (360 degrees) gives us a negative coterminal angle.

Positive coterminal angle: π/2 + 2π = 5π/2.

Negative coterminal angle: π/2 - 2π = -3π/2.

I think its b

Hope this helps

Hello!

The most logical answer would be a quiangular hexagon

I hope it helps!