MathDB

Equilateral Triangle on Sine Curve

Source:

8/8/2024

The graph of

y=C\sin x

, where

C>0

is a constant, is drawn on the interval

[0,\pi]

. Suppose that there exists a point

P

on the graph such that the triangle with vertices

(0,0)

,

(\pi,0)

, and

P

is equilateral. Find

C^2

.

trigonometry2022

Equiangular Octagon

Source:

8/8/2024

The side lengths of an equiangular octagon alternate between

20

and

22

. Find its area.

geometry2022

Derivative of Definite Integral

Source:

8/9/2024

Compute the value of

\frac{d}{dx}\int_{1}^{10} x^3\,dx.

calculusderivativeintegration2022

Spelling Johns Hopkins Correctly

Source:

8/8/2024

Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a

\frac{1}{3}

and

\frac{1}{4}

chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a

\frac{9}{10}

chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.

probability2022

Greatest Possible Coefficient Sum of Quartic Polynomial

Source:

8/8/2024

If three of the roots of the quartic polynomial

f(x) = x^4 + ax^3 + bx^2 + cx + d

are

0

,

2

, and

4

, and the sum of

a

,

b

, and

c

is at most

12

, then find the largest possible value of

f(1)

.

algebrapolynomial2022

1

Problems(5)