40 Questions
What is the equivalent of log(x) = 3?
ln(x) = 3
What is the base of natural logarithms?
e
In the given triangle ABC, what is true about the sides?
a = 2b
What can be said about angle C in triangle ABC?
cos(C) = -1/2
What is true about the circle with center O?
Segment AB is always a secant
What is true about quadrilateral ABCD?
∠𝐵 = ∠𝐷
What are the coordinates of the other endpoint?
(4, 5)
What is the equation of the line through (1,-1)?
x + 2y = 1
What is the name of a polygon with 40 sides?
tetracontagon
What is the value of p if the area of a rectangle decreases by 12% when its length is increased by 10% and its width is decreased by p%?
20
What is the name of a straight line segment whose endpoints lie on a circle?
chord
What is the solution to the equation Arctan x + Arctan (1 - x) = Arctan 4/3?
x = 1/2
What is the speed of the boat?
427.88 ft/min
What is the general form of the equation of a circle with center at (3,1) and tangent to the line 4x + 3y + 5 = 0?
x + y - 6x - 2y - 6 = 0
What is the shift of the graph 𝑦 = sin (𝑥 + )?
left 1 unit
What is the base edge of a regular triangular pyramid with an altitude of 9 m and a volume of 187.06 cu.m?
6
What is the approximate number of votes received by the first candidate in a city council election?
10,656
What is the equation of the perpendicular lines?
y = 3x + 2 and y = 3x − 2
What is the range of 𝑓(𝑥) = 2𝑥 − 6𝑥 + 7?
𝑓(𝑥) ≥ 7
What is the simplified form of the expression +?
3 cos A
What is the equation of the ellipse?
2𝑥 − 17𝑥 + 9 = 0
What is the height of the triangle?
(𝑥 − 2)
What is the volume of the cube?
67.43 cm3
What is the range of the radio equipment?
±3𝑖
What is the ratio of the volume of the sphere to the volume of the cylinder if a sphere is inscribed in a cylinder?
one-half
Can two ships, one 155 km N42 40′E and the other 165 km N45 10′W of a shore station, communicate directly?
No, they are 215.0347 km apart
What is the volume of water in a spherical tank having a diameter of 4 m?
5.236 m3
What is the depth of the water in a spherical tank having a diameter of 4 m and a volume of 5.236 m3?
1.0 m
What is the volume of a regular triangular prism with a base edge of 2 m circumscribing a sphere of radius 3 m?
9.59 m3
A right circular cone is cut into two parts of equal weight. Where should the cut be made?
0.595h
What is the volume of the portion of a sphere included between two parallel circular sections with radii 4 in. and 4.8 in.?
The volume can be calculated using the radii of the sections.
What is the total surface area of a right circular cone?
πr(r + l)
What is the volume of the right circular cone with base radius 10 cm and altitude 15 cm?
314.4 in3
What is the lateral surface area of a closed cylindrical tank with height 10 ft and diameter 4 ft?
384.52 cm2
How many iron rivets can be made from two kilograms of metal?
Cannot be determined
What is the height of the water in a horizontal cylindrical tank?
3.5 ft
What is the height of the water in a hexagonal prism-shaped flower vase?
34.88 in
What is the lateral area of a right section of an oblique prism with a hexagonal base?
428 cm3
What is the value of h'(x) given h(x) =
h'(x) = x + 1
What is the value of p for which (x - p)(x + 3 - p) has a remainder of 4?
5, 1
Study Notes
Equations and Logarithms
- Solving the equation log(x) = 3 is equivalent to solving ln(x) = 3, but with different numerical answers.
- ln(x) is equivalent to loge(x), where e is an irrational number approximately equal to 2.71828.
- Common logarithms (log) are typically used with a base of 10, while natural logarithms always use base e.
Geometry
- A chord is an extension of a chord in a circle which is a straight line segment of which the endpoints lie on the circle.
- A polygon with 40 sides is called a tetracontagon.
- In a quadrilateral ABCD, if AB = CD and ∠A = ∠C, then ∠B = ∠D.
Trigonometry
- sin(A) + sin(1 - A) = sin(4/3)
- Simplify + as 3cos(A) or 2sec(A)tan(A)
Analytic Geometry
- The general form of the equation of a circle with center at (3,1) and tangent to the line 4x+3y+5=0 is x + y - 6x - 2y - 6 = 0.
- The equation of an ellipse satisfying certain conditions is + =1 or + =1.
Three-Dimensional Geometry
- The volume of a sphere is that of a cylinder, and one-half that of a cylinder if the sphere is inscribed in the cylinder.
- If a sphere is inscribed in a cylinder, the volume of the sphere is one-half that of the cylinder.
Miscellaneous
- In a right triangle, the length of the leg opposite the 45-degree angle is the geometric mean of the hypotenuse and the leg adjacent to the 45-degree angle.
- If two ships have radio equipment with a range of 200 km, they can communicate directly if they are within 190.0234 km of each other.
I hope this helps! Let me know if you need further clarification.
Compare the solutions of logarithmic equations, including log(x) = 3 and ln(x) = 3, and identify the properties of their solution sets.
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