Mr Alexander built a rectangular fish tank. The length, breadth and height of the tank are 3x metres, 1.5 metres and (1 - x) metres respectively, as shown in the diagram below. Determine a formula for the volume of the tank in terms of x. Hence, determine the value of x that will maximise the volume of the tank. During an experiment, learners must record the velocity (v) of an electronic toy car over a distance (m), t seconds after the experiment has begun. The velocity of the electronic toy car is given by $v(t) = 8 + 4t - t^2$. Determine: The initial velocity of the toy car. The velocity of the toy car when t = 0.2 seconds. The rate at which the velocity changes with respect to time when t = 1.2 seconds.

NSC Technical Mathematics Mensuration: Mr Alexander built a rectangular

Mr Alexander built a rectangular fish tank - NSC Technical Mathematics - Question 8 - 2018 - Paper 1

Question 8

Mr Alexander built a rectangular fish tank. The length, breadth and height of the tank are 3x metres, 1.5 metres and (1 - x) metres respectively, as shown in the dia... show full transcript

Worked Solution & Example Answer:Mr Alexander built a rectangular fish tank - NSC Technical Mathematics - Question 8 - 2018 - Paper 1

Step 1

Determine a formula for the volume of the tank in terms of x.

96%

114 rated

Answer

To find the volume (V) of the rectangular tank, we use the formula for volume:

$V = l \times b \times h$

In our case, substituting the dimensions, we get:

V = (3x) \times (1.5) \times (1 - x) = 4.5x(1 - x) = 4.5x - 4.5x^2

Therefore, the formula for the volume of the tank in terms of x is: $V = 4.5x - 4.5x^2$

Step 2

Hence, determine the value of x that will maximise the volume of the tank.

99%

104 rated

Answer

To find the value of x that maximises V, we calculate the derivative of V with respect to x:

$rac{dV}{dx} = 4.5 - 9x$ Setting the derivative equal to zero to find critical points:

$4.5 - 9x = 0$ $9x = 4.5$ $x = 0.5$ Thus, the value of x that maximises the volume is: $x = 0.5$

Step 3

The initial velocity of the toy car.

96%

101 rated

Answer

To find the initial velocity of the toy car, we evaluate the function at t = 0:

$v(0) = 8 + 4(0) - (0)^2 = 8$ Thus, the initial velocity of the toy car is 8 m/s.

Step 4

The velocity of the toy car when t = 0.2 seconds.

98%

120 rated

Answer

To find the velocity at t = 0.2 seconds, we substitute t into the velocity function:

= 8 + 0.8 - 0.04 \newline = 8.76$$ The velocity of the toy car when t = 0.2 seconds is 8.76 m/s.

Step 5

The rate at which the velocity changes with respect to time when t = 1.2 seconds.

97%

117 rated

Answer

To find the rate of change of velocity, we first need the derivative of the velocity function:

$\frac{dv}{dt} = 4 - 2t$ Now, we evaluate this at t = 1.2 seconds:

$\frac{dv}{dt} |_{t=1.2} = 4 - 2(1.2) = 4 - 2.4 = 1.6$ The rate at which the velocity changes with respect to time when t = 1.2 seconds is 1.6 m/s².

Mr Alexander built a rectangular fish tank - NSC Technical Mathematics - Question 8 - 2018 - Paper 1

Worked Solution & Example Answer:Mr Alexander built a rectangular fish tank - NSC Technical Mathematics - Question 8 - 2018 - Paper 1

Determine a formula for the volume of the tank in terms of x.

Hence, determine the value of x that will maximise the volume of the tank.

The initial velocity of the toy car.

The velocity of the toy car when t = 0.2 seconds.

The rate at which the velocity changes with respect to time when t = 1.2 seconds.

Join the NSC students using SimpleStudy...