Alan

"A steep mountain is inclined 70 degree to the horizontal and rises 3500 ft above the surrounding plain. A cable car is to be installed by connecting a cable from the top of the mountain to a spot on the plain that is 1000 ft from the base of the mountain. Find the shortest length of cable needed. "

"What is the rectangular equivalence to the parametric equations?

x(θ)=4cosθ+2,y(θ)=2sinθ−5 , where 0≤θ<2π .

Drag a term into each box to correctly complete the rectangular equation."

The sum is 1.  See the following:

"Two circles of radius r are placed in a square with side length 2, without overlap  What is the largest possible value of r?"

Let r1 = fill rate of pipe 1, r2 = fill rate of pipe 2, V = reservoir volume.

r1 = V/(7/6)  = 6V/7   (1)

r2 = V/t        (2)        t = time for pipe 2 to fill reservoir by itself

r1 + r2 = V/(3/4) = 4V/3     (3)

Use (1) in (3)

6V/7 + r2 = 4V/3    so   r2 = 10V/21   (4)

Use (4) in (2)

10V/21 = V/t    t = 21/10   t = 2hrs 6mins

"The surface area of sphere A is 96% more than the surfacde area of sphere B.  The volume of sphere A is r% more than the volume of sphere B.  What is the value of r?"

"Andy spins a fair spinner numbered 1 - 5 and rolls a fair dice. What is the probability of spinning a multiple of 4 and rolling a factor of 15?"

Prob spin multiple of 4:  p1 = 1/5  since there is only one possibility, namely 4 itself.

Prob roll a factor pf 15:  p2 = 1/2  since the factors are 1, 3 and 5, half of the numbers on the die.

These are independent events, so probability of both together: p = p1*p2 = (1/5)*(1/2) = 1/10

"Find positive integers a, b, c such that a^3 + b^3 + c^3 = 2024."

a = 8, b = 8, c = 10

Assuming the overbar on 'a' just indicates it's a vector, and doesn't mean transpose, then:

"A glass is half full of pure alcohol and another glass twice the volume is one third full.  Both glasses are then filled to the top with water and mixed together.  What is the fraction of alcohol in the resulting mixture?"

Edited to correct mistake.  Thnaks to EP for pointing this out.