A small garden measures 7 ft by 14 ft. A uniform border around the garden increases the total area to 170 ft2. What is the width of the border?

7*14 + 170 = (7+2w)(14+2w)

Now solve for w

A small garden measures 10 ft by 7 ft. A uniform border around the garden increases the total area to 180 ft^2. What is the width of the border?

To find the width of the border, let's first calculate the area of the garden.

The garden measures 7 ft by 14 ft, so the area of the garden is 7 ft * 14 ft = 98 ft².

Now, we know that the total area, including the garden and the border, is 170 ft². So, we need to find the difference between the total area and the garden area to get the area of the border.

Area of the border = Total area - Garden area
Area of the border = 170 ft² - 98 ft² = 72 ft².

Since the border has a uniform width, we can assume that it has the same width on all four sides of the garden. Let's assume that the width of the border is x ft.

The length of the garden, including the border, would be 7 ft + 2x ft (since there are two borders - one on each side).
The width of the garden, including the border, would be 14 ft + 2x ft (again, two borders - one on the top and one at the bottom).

The area of the garden, including the border, can be calculated as length * width.
(7 ft + 2x ft) * (14 ft + 2x ft) = 170 ft².

Expanding the equation:
(98 ft² + 14x ft + 28x ft + 4x² ft²) = 170 ft².

Combining like terms:
4x² + 42x + 98 = 170.

Subtracting 170 from both sides:
4x² + 42x + 98 - 170 = 0.

Simplifying the equation:
4x² + 42x - 72 = 0.

Now, we can solve this quadratic equation to find the value of x.

Using factoring, we can rewrite the equation as:
(2x - 4)(2x + 18) = 0.

Setting each factor equal to zero gives two possible solutions:
2x - 4 = 0 or 2x + 18 = 0.

Solving for x in each equation:
2x - 4 = 0 => 2x = 4 => x = 2.
2x + 18 = 0 => 2x = -18 => x = -9.

Since the width cannot be negative, we disregard the solution -9.

Therefore, the width of the border is 2 ft.