Question

g the top and bottom margins of a poster are each 12 cm and the side margins are each 8 cm. if the area of printed material on the poster is fixed at 1536 cm2, find the dimensions of the poster with the smallest cmheight cm

Answer 1

To minimize the height of a poster with a fixed printed area of 1536 cm² and given margins, we consider a square printed area for minimal dimensions. Assuming a square, each side of the printed area is √1536 cm, yielding a smallest possible height for the poster of approximately 63.2 cm including the margins.

Step-by-step explanation:

To find the dimensions of a poster with the smallest height, we must consider the fixed area of printed material and the margins. The total area of the poster, which includes the printed area and margins, is given by:

Area of poster = (Width + Side Margins) * (Height + Top and Bottom Margins)

Given that the side margins are each 8 cm and the top and bottom margins are each 12 cm, we can express the dimensions of the whole poster as:

Width of poster = Width of printed material + 16 cm (8 cm on each side)

Height of poster = Height of printed material + 24 cm (12 cm on top and bottom)

The area of the printed material on the poster is 1536 cm². Therefore, the dimensions of the poster need to result in an area that includes this 1536 cm² plus the area taken up by the margins.

To find the smallest possible height, we need to minimize the Height of the poster regarding its fixed printed area.

Let's denote the Width of the printed material as x and the Height as y. So, the total area (A) of the poster is:

A = (x + 16)(y + 24)

But we know that the printed material area is x * y = 1536 cm². We now need to consider how to minimize y, considering this constraint. This is a problem that requires calculus, specifically minimizing a function with a constraint, which can be addressed using methods such as the derivative test or Lagrange multipliers.

However, without applying calculus, we can argue that a square will have the smallest perimeter for a given area, which usually extends to having the smallest total dimension sums for a fixed area. So the smallest height could be achieved when the printed area is formed as a square, meaning x = y.

If x = y, then we have:

x * x = 1536 cm²

x = √1536 cm²

x ≈ 39.2 cm (rounded to one decimal place)

Then the smallest height of the poster, which would be the height of the square printed area plus the top and bottom margins, can be calculated as:

Height of poster = 39.2 cm + 24 cm = 63.2 cm

So, the dimension of the poster for the smallest height would be approximately 63.2 cm in height (including the top and bottom margins).

Answer 2

The dimensions of the poster with the smallest perimeter can be found using calculus. The real-life area of the conference center's meeting room is 150,000 cm², based on the scale provided. Samir ran a 10,000 meters race corresponding to the 10 kilometers mentioned.

Step-by-step explanation:

The dimensions of the poster with the smallest perimeter that still meets the conditions given can be found by setting up an equation for the area of the printed material and using calculus to minimize the poster's overall area. The area of the printed material is 1536 cm², and since the top and bottom margins are 12 cm each and the side margins 8 cm each, the total dimensions of the poster would be the area of the printed material plus the margins. This leads to the formation of a function that can be differentiated to find the minimum size poster.

As for the conference center scale drawing, the actual area of the meeting room can be found by converting the scale measurements into actual measurements and then calculating the area. With the scale given as 1 cm : 2 m, a 1.5 cm by 2.5 cm drawing would correspond to an actual room size of 3 m by 5 m. The area in real-life measurements would be 3 m * 5 m = 15 m², and since each meter squared is 10,000 cm², the area would be 15 * 10,000 cm² = 150,000 cm².

For Samir's race, since 1 kilometer is equivalent to 1000 meters, running a 10-kilometer race means Samir ran 10 * 1000 meters, which is 10,000 meters.