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Hello,

To find the largest possible value of CF, we need to consider the relationship between altitudes and sides of a triangle. By using the formula CF = 2 * area(ABC) / AB, the largest possible value of CF occurs when AB is minimized. Therefore, CF will be maximum when AB is minimized to its smallest possible value.

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1.  The shortest path the gecko can take is a diagonal line across the room. The length of this diagonal line is the hypotenuse of a right triangle with legs of length 10 and 12. The length of the hypotenuse is given by the Pythagorean Theorem:

a^2 + b^2 = c^2

where a and b are the lengths of the legs and c is the length of the hypotenuse. Substituting in the lengths of the legs, we get:

10^2 + 12^2 = c^2

100 + 144 = c^2

244 = c^2

c = \sqrt{244} = 2\sqrt{61}

Therefore, the length of the shortest path the gecko can take is 2*sqrt(61)​​ feet.

Since 5 < n/k < 6, we know that k is a divisor of n but k < n < 6k. Therefore, the smallest possible value of k is 2 and the smallest possible value of n is 11. In this case, lcm(n,k) = 22 and gcd(n,k) = 2, so lcm(n,k)/gcd(n,k) = 11. Therefore, the smallest possible value of lcm(n,k)/gcd(n,k) is 11.

wait no i did the calulations and i got 22,

@melody pls tell me if im wrong

To find the measure of minor arc RS, we need to determine the measure of angle SOT first.

We are given that angle RTB measures 28 degrees. Since angle RTB is an inscribed angle, its measure is half the measure of the intercepted arc RS. Therefore, the measure of intercepted arc RS is 2 * 28 = 56 degrees.

Now, let's denote the measure of angle SOT as x degrees. According to the given information, the measure of angle ROB is three times the measure of angle SOT. Therefore, angle ROB measures 3x degrees.

Since angle ROB is also an inscribed angle, its measure is half the measure of the intercepted arc RS. Thus, we have:

3x = 56 x = 56 / 3

Finally, the measure of minor arc RS is twice the measure of angle SOT, since it is intercepted by angle RTB. Thus:.

2 * 56/3 = 112/3 degrees

Therefore, the measure of minor arc RS is 112/3 degrees.

Right scalene triangle.

Sides: a = 20   b = 17.321   c = 10

Area: T = 86.603
Perimeter: p = 47.321
Semi-perimeter: s = 23.66

Angle ∠ A = α = 90° = 1.571 rad
Angle ∠ B = β = 60° = 1.047 rad
Angle ∠ C = γ = 30° = 0.524 rad

Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.

Inradius ==Area / semi-perimeter ==86.603 / 23.66 ==3.66

Since AC is the diameter of the semicircle, O is the center of the semicircle. Since AB is a diameter of the semicircle, ∠AOB=90∘. Therefore, triangle AOB is a right triangle with legs of length 10 and 12, so its area is (1/2)(10)(12)=60. The area of the semicircle is (1/2)(202)(pi)=200pi. The area of the shaded region is the difference between the area of the semicircle and the area of triangle AOB, or 200pi−60=140pi−60​.

Therefore, a + b = 140 + 60 = 200.

Try: sqrt(32 + sqrt(2)) ==sqrt(33.14142) ==5.7805

We know that gcd(n,k) divides both n and k, so gcd(n,k)n​ and gcd(n,k)k​ are integers. Also, lcm(n,k) is a multiple of both n and k, so gcd(n,k)lcm(n,k)​ is an integer. Therefore, [\frac{\text{lcm}(n,k)}{\gcd(n,k)} = \frac{n}{\gcd(n,k)} \cdot \frac{k}{\gcd(n,k)}.]We are given that 5

We know that AB is a diagonal of the larger square, so AB=6*sqrt(2)​. Also, WP is a diagonal of one of the small squares, so WP=2*sqrt(2)​. Therefore, the maximum possible distance from A to P is AB−WP=4*sqrt(2)​​.

The previous person also gave this answer but it incorrect.

Hello! Can you please share the diagram?

You get sqrt(34) by rotating square ABCD so that the vertices were in the center of the 2x2 squares. Then, calling the midpoint of the bottom line E, you get right triangle AEP. AE = 5, EP = 3, so AP = sqrt(34). 

Hello! Here's the link to the solution: https://web2.0calc.com/questions/helpelp#r3. PLEASE let me know if the answer is wrong because I noticed this is not the first time you posted this and the other person direct the conversation to the link above.

Hello!

a) the volume, answer, is 96 in^2

The volume of a prism formula is v = bh. Since we know both the base and the height, we can easily find the volume. So the volume is 8in*12in = 96 in^2.

b) Im unsure about this part because i dont know what type of prism it is. It would be different for a triangular prism and a rectangular prism so... Perhaps someone else can help with this part.

Nope, can you please help? I'll be active again in couple of hours. I need to take a break from math; I have been doing math for 3 hours now.

Correct! Here's the full and detailed solution. Let me know if you have any questions!

Thomas can choose any \(4\) out of \(12\) players, which is \(\binom{12}{4}\) distinct possibilities for a team. Carrie can choose any \(4\) out of the remaining \(8\) players, which is \(\binom{8}{4}\) distinct possibilities for a team. Lenny has only \(1\) choice for his team, whichever \(4\) have not yet been chosen. Combining all this yields

\(\begin{align*} \frac{12\times11\times10\times9}{4\times3\times2}\times\frac{8\times7\times6\times5}{4\times3\times2} &= 11\times 10 \times 9 \times 7 \times 5 \\ &= 99\times 35\times 10 \\ &= (3500-35)\times 10 \\ &= \boxed{34,650\text{ ways}}. \end{align*}\)

Do you know the answer to this one?

[12 nCr 4] * [8 nCr 4] * [4 nCr 4]==34,650 - different ways of picking the 3 teams.

TYSM! Yes, I do understand the solution.

Add. Final answer needs to be in Standard Form. (−7x − x 3 + 26x 2 − 13) + (8x − x 3 + 6x 4 + 27)

 does that help??

See very detailed answer #2 here. I hope you understand it:

https://web2.0calc.com/questions/the-powers-of-coffee-in-mathematicians#r2

Incorrect solution again... The answer is 34,650.

Thomas can choose from 12 players to form his team, Carrie can choose from the 8 remaining players, and Lenny can choose from the 5 remaining players. Therefore, there are 12×8×5=480​ ways to form the teams.

Incorrect, the answer is 6 choose 4 which is 15.