A park in the shape of a rectangle has dimensions 60m by 100m. If the park contains a rectangular garden enclosed by a concrete terrace, how wide is the terrace if the are of the garden is one-half the area of the park?

park area: 60*100 = 6000

If the terrace has uniform width w, then
garden area is (60-2w)(100-2w)

(60-2w)(100-2w) = 3000

Now just solve for w.

Well, it seems like the garden wants to shake things up by taking up only half the area of the park! But don't worry, I'm here to help. Let's solve this mystery.

The area of the park is 60m * 100m = 6000 square meters.

Since the garden is one-half the area of the park, its area would be 6000 / 2 = 3000 square meters.

Now, let's find the dimensions of the garden. Let's say the width of the garden is 'x.'

The length of the garden would be the total length of the park (60m) minus twice the width of the terrace because there's a terrace on both sides.

So, the length of the garden would be 60m - 2x.

The area of the garden is the product of its length and width, which gives us the equation 3000 = (60m - 2x) * x.

Simplifying the equation, we have 3000 = 60x - 2x^2.

Now, let's rearrange it and set it equal to zero: 2x^2 - 60x + 3000 = 0.

But wait! We can divide everything by 2 to make our life a bit easier: x^2 - 30x + 1500 = 0.

Now, let's put our detective hats on and solve this quadratic equation.

Unfortunately, my humor won't be much help here, so we'll turn to the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a.

For our equation, a = 1, b = -30, and c = 1500.

Calculating it, we find two possible values for x: x ≈ 25.83 or x ≈ 4.17.

Since we're talking about dimensions here, we'll go with the positive value: x ≈ 4.17 meters.

So, the width of the terrace would be approximately 4.17 meters.

Voila! Mystery solved!

To find the width of the terrace, we need to first calculate the area of the garden and then subtract it from the total area of the park.

Given that the dimensions of the park are 60m by 100m, the total area of the park is calculated by multiplying the length and width:

Area of the park = length x width = 60m x 100m = 6000m²

Next, we need to find the area of the garden. It is given that the area of the garden is half the area of the park, so we divide the area of the park by 2 to find the area of the garden:

Area of the garden = Area of the park / 2 = 6000m² / 2 = 3000m²

Since the garden is in the shape of a rectangle, we can use the formula for the area of a rectangle to find its dimensions.

Let's assume the length of the garden is L and the width of the garden is W. Then the area of the garden can be calculated as:

3000m² = L x W

Now, we need to find the dimensions of the garden. Solving for W, we divide both sides of the equation by L:

W = 3000m² / L

Since we know the shape of the park is a rectangle with dimensions 60m by 100m, we can assume that the garden is also a rectangle with the same length (60m) or width (100m) as the park.

If we assume that L = 60m, then the width of the garden (W) can be calculated as:

W = 3000m² / 60m = 50m

Similarly, if we assume that L = 100m, then the width of the garden (W) can be calculated as:

W = 3000m² / 100m = 30m

Therefore, the width of the terrace can be calculated by subtracting the width of the garden from the width of the park.

If L = 60m, then the width of the terrace is 100m - 50m = 50m.

If L = 100m, then the width of the terrace is 60m - 30m = 30m.

So, the width of the terrace could either be 50m or 30m, depending on the length of the garden.

To find the width of the terrace, we first need to find the area of the garden and the park.

The area of the park can be calculated by multiplying its length (60m) by its width (100m):

Area of the park = 60m × 100m = 6000 square meters.

Since the area of the garden is given to be one-half the area of the park, we can find the area of the garden by dividing the area of the park by 2:

Area of the garden = 6000 square meters ÷ 2 = 3000 square meters.

Now, let's assume the width of the terrace as 'x' meters. This means that the dimensions of the garden would be reduced by twice the width of the terrace (2x), resulting in a smaller rectangle within the park.

Therefore, the length of the garden would be reduced to (60m - 2x) and the width would be reduced to (100m - 2x).

The area of the garden can also be calculated using its dimensions:

Area of the garden = (60m - 2x) × (100m - 2x).

We know that the area of the garden is equal to 3000 square meters, so we can set up the equation:

(60m - 2x) × (100m - 2x) = 3000 square meters.

Now, let's solve the equation to find the width of the terrace:

(60m × 100m) - (120m × x) - (200m × x) + 4x² = 3000.

6000 - 320x + 4x² = 3000.

4x² - 320x + 3000 = 0.

Dividing the equation by 4 to simplify it:

x² - 80x + 750 = 0.

Now, we can either factor the quadratic equation or use the quadratic formula to find the value of 'x'. Factoring the equation, we get:

(x - 50) (x - 30) = 0.

This gives us two possible solutions: x = 50 or x = 30.

However, since we're looking for the width of the terrace, the value of 'x' cannot be equal to the length or width of the park.

Therefore, the width of the terrace is 30 meters.