consider the distance equation d=rt, where d is the distance (in feet), r is the rate (in feet per second), and t is the time (i
1 answer:
Answer:
yes, they are proportional
Step-by-step explanation:
A proportion is a relation that can be modeled by a straight line through the origin. The equation will have the form ...
y = kx
where x is the independent variable, y is the dependent variable, and k is a constant of proportionality.
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If you start with the equation relating distance, rate, and time:
d = rt
and you fix the time at 50 seconds, then the equation becomes ...
d = 50t
This is the equation of a proportion with a constant of proportionality of 50. It tells you the distance run is proportional to the rate you run. When this equation is graphed, it is a straight line through the origin.
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You know that 
so 
.
Area of a shaded region is:
![$A=\int\limits_0^k\dfrac{y^2}{4}\,dy=\frac{1}{4}\int\limits_0^ky^2\,dy=\frac{1}{4}\left[\frac{y^3}{3}\right]_0^k=\frac{1}{4}\left[\frac{k^3}{3}-\frac{0^3}{3}\right]=\frac{1}{4}\cdot\dfrac{k^3}{3}=\boxed{\frac{k^3}{12}}](https://tex.z-dn.net/?f=%24A%3D%5Cint%5Climits_0%5Ek%5Cdfrac%7By%5E2%7D%7B4%7D%5C%2Cdy%3D%5Cfrac%7B1%7D%7B4%7D%5Cint%5Climits_0%5Eky%5E2%5C%2Cdy%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft%5B%5Cfrac%7By%5E3%7D%7B3%7D%5Cright%5D_0%5Ek%3D%5Cfrac%7B1%7D%7B4%7D%5Cleft%5B%5Cfrac%7Bk%5E3%7D%7B3%7D-%5Cfrac%7B0%5E3%7D%7B3%7D%5Cright%5D%3D%5Cfrac%7B1%7D%7B4%7D%5Ccdot%5Cdfrac%7Bk%5E3%7D%7B3%7D%3D%5Cboxed%7B%5Cfrac%7Bk%5E3%7D%7B12%7D%7D%20)
so k:
![\dfrac{k^3}{12}=\dfrac{9}{4}\\\\\\k^3=\dfrac{9\cdot12}{4}\\\\\\k^3=9\cdot3\\\\k^3=3^3\quad|\sqrt[3]{(\ldots)}\\\\\boxed{k=3}](https://tex.z-dn.net/?f=%5Cdfrac%7Bk%5E3%7D%7B12%7D%3D%5Cdfrac%7B9%7D%7B4%7D%5C%5C%5C%5C%5C%5Ck%5E3%3D%5Cdfrac%7B9%5Ccdot12%7D%7B4%7D%5C%5C%5C%5C%5C%5Ck%5E3%3D9%5Ccdot3%5C%5C%5C%5Ck%5E3%3D3%5E3%5Cquad%7C%5Csqrt%5B3%5D%7B%28%5Cldots%29%7D%5C%5C%5C%5C%5Cboxed%7Bk%3D3%7D)
Area of a shaded region is:
so k: