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Population growth rate - Logistic equation

In biology or human geography, population growth is the increase in the number of individuals in a population.

The “population growth ... more

Weighted average cost of capital

The weighted average cost of capital is the rate that a company is expected to pay on average to all its security holders to finance its assets. It is the ... more

Logarithmic Standard Deviation - 2nd moment

Shows how much variation or dispersion from the average exists. Logarithmic mean size (1st moment) needs to be precalculated.

... more

Worksheet 334

In a video game design, a map shows the location of other characters relative to the player, who is situated at the origin, and the direction they are facing. A character currently shows on the map at coordinates (-3, 5). If the player rotates counterclockwise by 20 degrees, then the objects in the map will correspondingly rotate 20 degrees clockwise. Find the new coordinates of the character.

To rotate the position of the character, we can imagine it as a point on a circle, and we will change the angle of the point by 20 degrees. To do so, we first need to find the radius of this circle and the original angle.

Drawing a right triangle inside the circle, we can find the radius using the Pythagorean Theorem:

Pythagorean theorem (right triangle)

To find the angle, we need to decide first if we are going to find the acute angle of the triangle, the reference angle, or if we are going to find the angle measured in standard position. While either approach will work, in this case we will do the latter. By applying the cosine function and using our given information we get

Cosine function

While there are two angles that have this cosine value, the angle of 120.964 degrees is in the second quadrant as desired, so it is the angle we were looking for.

Rotating the point clockwise by 20 degrees, the angle of the point will decrease to 100.964 degrees. We can then evaluate the coordinates of the rotated point

For x axis:

Cosine function

For y axis:

Sine function

The coordinates of the character on the rotated map will be (-1.109, 5.725)

Reference : PreCalculus: An Investigation of Functions,Edition 1.4 © 2014 David Lippman and Melonie Rasmussen
Creative Commons License : http://creativecommons.org/licenses/by-sa/3.0/us/


The G-measure is the geometric mean of Recall and Precision. Precision (also called positive predictive value) is the fraction of retrieved instances that ... more

Optical depth

In physics, optical depth or optical thickness, is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and ... more

F1 score

In statistical analysis of binary classification, the F1 score (also F-score or F-measure) is a measure of a test’s accuracy. It considers both the ... more

Vertical Curve - Parabolic formula

Vertical Curves are the second of the two important transition elements in geometric design for highways, the first being Horizontal Curves. A vertical ... more

Gravity measurement with seconds pendulum (Kater's pendulum))

A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a ... more

Worksheet 302

In the wheelbarrow of the following figure the load has a perpendicular lever arm of 7.50 cm, while the hands have a perpendicular lever arm of 1.02 m.(a) What upward force must you exert to support the wheelbarrow and its load if their combined mass is 45.0 kg? (b) What force does the wheelbarrow exert on the ground?

(a) In the case of the wheelbarrow, the output force or load is between the pivot and the input force. The pivot is the wheel’s axle. Here, the output force is greater than the input force. Thus, a wheelbarrow enables you to lift much heavier loads than you could with your body alone. (b) In the case of the shovel, the input force is between the pivot and the load, but the input lever arm is shorter than the output lever arm. The pivot is at the handle held by the right hand. Here, the output force (supporting the shovel’s load) is less than the input force (from the hand nearest the load), because the input is exerted closer to the pivot than is the output.


Here, we use the concept of mechanical advantage.

Force (Newton's second law)
Mechanical Advantage - Law of Lever

An even longer handle would reduce the force needed to lift the load. The MA here is:


Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.
Creative Commons License : http://creativecommons.org/licenses/by/3.0/

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