Daniela is a stain glass artist; she created the window below for her living room. What is

Mathematics

1 answer:

OLga [1]3 years ago

3 0

The window has a shape of a semi-circle (half of a circle).

We know the diameter d = 12 ft.

The area of a (complete) circle is given by the formula:
A = π × r²

The radius can be found by halving the diameter:
r = d ÷ 2
= 12 ÷ 2
= 6 ft

Therefore the area of the circle will be:
A = π × r²
= 3.14 × 6²
= 3.14 × 36
= 113.04 ft²

And the area of the semi-circle:
A = 113.04 ÷ 2
= 56.52 ft²

Hence, the area of the window will be 56.52 ft².

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miskamm [114]

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2 years ago

What’s the domain and range of: log(√(2x-1) + 3 ) Please explain how you got it too!!

Radda [10]

Two main facts are needed here:

1. The logarithm $\log x$ , regardless of the base of the logarithm, exists for $x>0$ .

2. The square root $\sqrt x$ exists for $x\ge0$ .

(in both cases we're assuming real-valued functions only)

By (2) we know that $\sqrt{2x-1}$ exists if $2x-1\ge0$ , or $x\ge\dfrac12$ .

By (1), we know that $\log(\sqrt{2x-1}+3)$ exists if $\sqrt{2x-1}+3>0$ , or $\sqrt{2x-1}>-3$ . But as long as the square root exists, it will always be positive, so this condition will always be met.

Ultimately, then, we only require $x\ge\dfrac12$ , so the function has domain $\left[\dfrac12,\infty)$ .

To determine the range, we need to know that, in their respective domains, $\sqrt x$ and $\log x$ increase monotonically without bound. We also know that $x=\dfrac12$ at minimum, at which point the square root term vanishes, so the least value the function takes on is $\log3$ . Then its range would be $[\log3,\infty)$ .