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Math Knowledge

Which of the following is not required to define the slope-intercept equation for a line?

x-intercept

A line on the coordinate grid can be defined by a slope-intercept equation: y = mx + b. For a given value of x, the value of y can be determined given the slope (m) and y-intercept (b) of the line. The slope of a line is change in y over change in x, \({\Delta y \over \Delta x}\), and the y-intercept is the y-coordinate where the line crosses the vertical y-axis.

Shop Information

A joint created by welding is __________ the original metal.

as strong or stronger than

The joint created by welding is as strong or stronger than the original metal.

Arithmetic Reasoning

What is \( \frac{1}{8} \) x \( \frac{1}{5} \)?

\(\frac{1}{40}\)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{1}{8} \) x \( \frac{1}{5} \) = \( \frac{1 x 1}{8 x 5} \) = \( \frac{1}{40} \) = \(\frac{1}{40}\)

Word Knowledge

The hedonism of the emperoro, who spent all his time giving parties, caused his downfall.

sensualism

Belief that pleasure is the sole aim of life.

Math Knowledge

Find the value of c:
5c + z = -1
-2c - 3z = -3

-\(\frac{6}{13}\)

You need to find the value of c so solve the first equation in terms of z:

5c + z = -1
z = -1 - 5c

then substitute the result (-1 - 5c) into the second equation:

-2c - 3(-1 - 5c) = -3
-2c + (-3 x -1) + (-3 x -5c) = -3
-2c + 3 + 15c = -3
-2c + 15c = -3 - 3
13c = -6
c = \( \frac{-6}{13} \)
c = -\(\frac{6}{13}\)

Arithmetic Reasoning

What is \( \frac{3}{3} \) - \( \frac{5}{5} \)?

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 5 are [5, 10, 15, 20, 25, 30, 35, 40, 45, 50]. The first few multiples they share are [15, 30, 45, 60, 75] making 15 the smallest multiple 3 and 5 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{3 x 5}{3 x 5} \) - \( \frac{5 x 3}{5 x 3} \)

\( \frac{15}{15} \) - \( \frac{15}{15} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{15 - 15}{15} \) = \( \frac{0}{15} \) =

Mechanical Comprehension

A = 5 ft., the green box weighs 50 lbs., and the blue box weighs 80 lbs. What does distance B need to be for this lever to balance?

3.13 ft.

In order for this lever to balance, the torque acting on side A must equal the torque acting on side B. Torque is weight x distance from the fulcrum which means that the following must be true for the lever to balance:

f_Ad_A = f_Bd_B

For this problem, the equation becomes:

50 lbs. x 5 ft. = 80 lbs. x d_B

d_B = \( \frac{50 \times 5 ft⋅lb}{80 lbs.} \) = \( \frac{250 ft⋅lb}{80 lbs.} \) = 3.13 ft.

General Science

"For every action, there is an equal and opposite reaction." This describes which of Newton's laws of motion?

third

Newton's third law of motion states that For every action, there is an equal and opposite reaction. When an object exerts a force on another object, the second object exerts a force of equal magnitude in the opposite direction on the first object.

Shop Information

This tool is a(n) __________.

punch

A punch is narrow and is used to drive objects like nails (pin punch) or for making guide marks for drilling (center punch) or patterns in wood or metal.

Mechanical Comprehension

The standard unit of energy is the:

Joule

The Joule (J) is the standard unit of energy and has the unit \({kg \times m^2} \over s^2\).