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i believe the equation is : 2 x pi x rso if the radius is half the diameter, it will be 2 x pi x 2.5 = 15.72 x pi x r is the circumfrence.. pi x r2= A so 2.52 x pi=6.25pi or about 19.63
Density is equal to the mass divided by the volume. Usually density is represented by the Greek letter rho, however here I will just use the letter "r". So then:r = M/V and thereforeM = rV and thereforeV = M/rSo if you know any two of r, M, or V, you can find the missing piece of information.
The area increases by 21%.Let r=the original radiusand let R=the increased radius10% of r is .1rR=r + .1 rso R=1.1ra=the original areaand A=the area of the larger circlea= pi x r2A= pi x R2=pi x (1.1r)2=pi x 1.21r2A=1.21 x (pi x r2)=1.21 aA=a + .21 aThe area of the new circle increases by 21%
I assume you mean covariance matrix and I assume that you are familiar with the definition:C = E[(X-u)(X-u)T]where X is a random vector and u = E(X) is the meanThe definition of non-negative definite is:xTCx ≥ 0 for any vector x Є RSo is xTE[(X-u)(X-u)T]x ≥ 0?Then, from one of the covariance properties:E[(xT(X-u))((X-u)Tx)] = E[xT([(X-u)(X-u)T]x)] = E[((xTI)x)] = E[xTx]Finally, since we've already defined x to have only real values, xTx is therefore non-negative definite by definition.