Other Math Answers

Modeling refers to the solve problems. Give at least three benefits that you may achieve in your primary mathematics class by employing modeling.

Amy and Bill are fixing up their house by painting walls and installing electrical outlets.



In one hour, Amy can paint 8 walls, or install 6 outlets.



In one hour, Bill can paint 5 walls, or install 5 outlets

If Bill painted 5 walls in the first hour of work, and then switched to doing 5 outlets in the second hour of work, what was the opportunity cost of each of those outlets?



Instructions: Answer in whole numbers.



For the third hour, if he switches back to painting 5 walls, what was the opportunity cost of each of those walls?

Mothers and pre-school teachers often boasted that a four-year old child could already
read numbers. Is this an ideal situation?

. (a) A number of particles with masses m1, m2, m3,··· ,mn are situated at the points with position vectors r1, r2, r3,··· ,rn relative to an origin O. The center of mass G of the particles is defined to be the point of space with position vector
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn .
Show that if a different origin O were used, this definition would still place G at the same point of space.

(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.

i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.

ii. Find the angle between F1 and F4.

. (a) A number of particles with masses m1, m2, m3,··· ,mn are situated at the points with position vectors r1, r2, r3,··· ,rn relative to an origin O. The center of mass G of the particles is defined to be the point of space with position vector
R= m1r1+m2r2+m3r3+···+mnrn /m1 +m2+m3+···+mn . Show that if a different origin Owere used, this definition would still place G at the same point of space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to the nearest Newton.
ii. Find the angle between F1 and F4.

Show that the geometric mean between x and y is ±√xy and the common ratio r is r=^n+1√y/a

The sum of five numbers in arithmetic progression is 25 and of their square is 165. Find the numbers.

The sum of the first n terms of a series is 2n^2 -2. Find the nth term and show that the series is an arithmetic progression.

A number of particles with masses m1, m2, m3, · · · , mn are situated at the points with position vectors
r1, r2, r3, · · · , rn relative to an origin O. The center of mass G of the particles is defined to be the
point of space with position vector
R =
m1r1 + m2r2 + m3r3 + · · · + mnrn
m1 + m2 + m3 + · · · + mn
.
Show that if a different origin O0 were used, this definition would still place G at the same point of
space.
(b) An object of mass 40kg is supported in equilibrium by four cables. The forces, in Newtons, exerted
by three of the cables, F1, F2 and F3, are given in terms of the unit vectors, i, j and k as F1 =
80i + 20j + 100k, F2 = 60i − 40j + 80k and F3 = −50i − 100j + 80k. The unit vectors i and j are
perpendicular and horizontal and the unit vector k is vertically upwards.
i. Find F4, the force exerted by the fourth cable, in terms of i, j and k. Also find its magnitude to
the nearest Newton.
ii. Find the angle between F1 and F4.

The position vector r of a particle at time t is
r = a cos (ωt)i + b sin (ωt)j
where a, b, ∈ R and a 6= b.
(a) Find the velocity and acceleration vectors v and a respectively.
(b) Show that
ω
2
|v|
2 + |a|
2 = ω
4
(a
2 + b
2
),
and find the times at which the velocity and the acceleration vectors are perpendicular.