Answer: Its light-collecting area would be 25 times greater than that of the 10-meter keck telescope.
Step-by-step explanation:
1. To solve this problem you must apply the formula for calculate the area of a circle, which is shown below:
[tex]A=r^{2}\pi[/tex]
Where r is the radius of the circle.
2. The diameter of a circle is:
[tex]D=\frac{r}{2}[/tex]
Where r is the radius of the circle.
3. Therefore, keeping this on mind, you have that the light-collecting area of a 50-meter keck telescope is:
[tex]A_1=(\frac{50m}{2})^{2}\pi=1963,49m^{2}[/tex]
4. And the light-collecting area of a 10-meter keck telescope is:
[tex]A_2=(\frac{10m}{2})^{2}\pi=78.53m^{2}[/tex]
5. Divide [tex]A_1[/tex] by [tex]A_2[/tex], then:
[tex]\frac{1963.49m^{2}}{78.53m^{2}}=25[/tex]
6. Therefore, its light-collecting area would be 25 times greater than that of the 10-meter keck telescope.
The given diameter of the telescopes of 50-meters and 10-meter, give
the number of times greater the light collecting area will be as 25 times.
The light collecting diameter of a 50-meter telescope = 50 m.
The light collecting diameter of a 10-meter telescope = 10 m.
The light collecting area of a telescope = [tex]\pi \cdot \frac{D^2}{4}[/tex]
Which gives;
[tex]Ratio \ of \ light \ collecting \ area = \dfrac{\pi \cdot \dfrac{\left(50 \, m\right)^2}{4}}{\pi \cdot \dfrac{\left(10 \, m \right)^2}{4}} = \dfrac{2,500}{100} = \mathbf{ 25}[/tex]
Learn more about finding the area of a circle here:
https://brainly.com/question/362136
