GYanggg

Assuming you already have the diagram, here is what I got:

Since angle EDF is an inscribed angle, we have:

\(\angle{EDF}=\frac{{\text{arc}\ EF}}{2} \)

Since angle A is formed by two tangents, we find:

\(\angle{A}=(\text{arc}\ EDF - \text{arc}\ EF)/2=(360º-2\cdot\text{arc}\ EF)/2\)

Therefore, \(\angle{A}=180º-\text{arc}\ EF = 180º - 2 \angle{EDF}\)

Solving for EDF, we get \(\angle{EDF}=90º-{\angle{A}}/2\)

Since we know angle A is greater than zero, angle EDF is less than 90. Similarly, other angles are also acute. 

I hope this helped,

Gavin

Hi ant101!

The sum of eleven consecutive integers is 11. What is the least of these eleven integers?

Since there are 11 consecutive integers, we can assign a variable to one of the numbers, then write all the other numbers in terms on the variable.

Since there are an odd number of integers, we can call the middle one n.

The five numbers greater than n are in order from least to greatest: \(n+1, n+2, n+3, n+4, n+5\)

Also, the five numbers less than n are in order from least to greatest: \(n-5, n-4, n-3, n-2, n-1\)

Adding them up, we have:

\((n-5)+(n-4)+(n-3)+(n-2)+(n-1)+n+(n+1)+(n+2)+(n+3)+(n+4)+(n+5)=11n.\)

Therefore the sum of the eleven integers can be expressed as 11n. 

As you said, the sum of these integers is 11, so we have:

\(11n=11\\ n=1\)

The least of these integers is \(n-5, \text{so} \ 1-5=\boxed{-4}\)

I hope this helped,

Gavin

Hey Hev!

We can see this is an arithmetic sequence, 

It is obvious because the formula in the summation notation gives us the formula, 3n - 1 

We just need to find the sum of the 8 terms. 

To find the sum of an arithmetic sequence, we use the formula:

\(S_n=\frac{n}2(a_1+a_n)\), where \(S_n\) is the sum, n is the number of terms, a₁ is the first term, and a_n is the last term.

We plug in the values of n into the formula, to find the term value. 

\(S_n=\frac82(2+23)=100\), that is your final answer. 

I hope this helped,

Gavin

Hey Hev!

When n = 1,

\(4(-5)^{1-1}=4(-5)^0=4\cdot1=4\)

When n = 2, 

\(4(-5)^{2-1}=4(-5)^1=4\cdot-5=-20\)

When n = 3,

\(4(-5)^{3-1}=4(-5)^2=4\cdot25=100\)

We can tell there is a pattern with multiplying by -5. 

The next three terms in order are -500, 2500, and -12500

You add them to find the solution,

I hope this helped,

Gavin

Hi QS!

Since the two shapes are similar, we can set up the following relationship:

Assumption 1: triangle ABE is similar to ACE, CD and BE is parallel 

\(\frac{15}{15+3}=\frac{10}{10+x}\\ 15(10+x)=10\cdot{18}\\ 150+15x=180\\ 15x=30\\ x=2\)

Assumption 2: triangle ABE is similar to ACE, CD and BE are not paralle

\(\frac{10}{15+3}=\frac{15}{10+x}\\ 10(10+x)=15\cdot18\\ 100+10x=270\\ 10x=170\\ x=17\)

I hope this helped,

Gavin

Hi QS!

Since the two shapes are similar, we can set up the following relationship:

Assumption 1: triangle ABE is similar to ACE, CD and BE is parallel 

\(\frac{15}{15+3}=\frac{10}{10+x}\\ 15(10+x)=10\cdot{18}\\ 150+15x=180\\ 15x=30\\ x=2\)

Assumption 2: triangle ABE is similar to ACE, CD and BE are not paralle

\(\frac{10}{15+3}=\frac{15}{10+x}\\ 10(10+x)=15\cdot18\\ 100+10x=270\\ 10x=170\\ x=17\)

I hope this helped,

Gavin

I understand you tertre, but this is just one less step. 

We don't actually need to calculate the number of brownies, but just the percentage. 

Ava stacked 0.14 of the brownies, this is 14%. 

Grace stacked 4/50 brownies, which is 8%.

Jayden stacked 24%. 

Zoe stacked 100 - 14 - 8 - 24 = 54%

Therefore, Zoe stacked the most. 

There is a more reliable way to do this. 

We notice that these can be expressed as points of a line,

\((1,5);(2,9);(3,13)...\)

We can treat this as a function problem, and we need to solve for the equation of the line. 

You first find the slope. 

\(\frac{9-5}{2-1}=4\\ y=4x+b\\ 5 = 4\cdot1+b\\ b=1\)

The equation of the line is \(y=4x+1\)

Now we plug in x = 10, 

\(y = 4\cdot10+1=\boxed{41}\)

I hope this helped,

Gavin

Hello Hev123!

The nth term of an arithmetic sequence with the first term A and common difference B, can be expressed as:

\(a_n=A+B(n-1)\)

The explicit rule given is in simplified form, so we need to find the first term and common difference. 

We plug 1 for n, in the given rule, and we get:

\(a_1=48-11\cdot1=37\)

We know the first term in 37. To find the common difference, we want to find the second term.

\(a_2=48-11\cdot2=26\)

The common difference is 26 - 37 = -11. 

You are correct.

I hope this helped,

Gavin