answer：First, I recognize that 15% of the population is affected by the disease.Next, 8% of these affected individuals die from the disease.To find the overall mortality percentage in relation to the entire population, I multiply the percentage of affected individuals by the mortality rate among them: 15% multiplied by 8% equals 1.2%.Therefore, the percentage of mortality in relation to the entire population is 1.2%.

answer：To determine how many prime numbers between 30 and 65 have a prime remainder when divided by 10, I will follow these steps:First, I will list all the prime numbers within the range of 30 to 65. These primes are 31, 37, 41, 43, 47, 53, 59, 61, and 67.Next, for each of these primes, I will divide them by 10 and identify the remainder. The possible remainders when dividing by 10 are 1, 3, 7, and 9.Then, I will check if each of these remainders is a prime number. The prime remainders are 3, 7, and 9 is not prime.Finally, I will count how many of the original primes have a prime remainder when divided by 10. The primes 37, 43, 47, 53, 59, and 67 meet this criterion, resulting in a total of 6 primes.

answer：To determine when the expressions ( a + bc ) and ( (a + b)(a + c) ) are equal, I'll start by expanding the second expression.Expanding ( (a + b)(a + c) ) gives:[(a + b)(a + c) = a^2 + ac + ab + bc]Next, I'll set the two expressions equal to each other:[a + bc = a^2 + ac + ab + bc]Subtracting ( a + bc ) from both sides to simplify:[0 = a^2 + ac + ab - a]Factoring out ( a ) from the right-hand side:[0 = a(a + c + b - 1)]This equation implies that either ( a = 0 ) or ( a + b + c - 1 = 0 ).If ( a = 0 ), substituting back into the original expressions shows they are equal:[0 + bc = (0 + b)(0 + c) implies bc = bc]If ( a + b + c - 1 = 0 ), then ( a + b + c = 1 ), and substituting this into the expressions also shows they are equal.Therefore, the expressions are equal whenever ( a + b + c = 1 ).

answer：Okay, so I have this problem here where I need to find the value of ( a + b ) given the function ( f(x) = log_a(x + b) ). The graph of this function passes through the point ( (2, 1) ), and the graph of its inverse function passes through the point ( (2, 8) ). Hmm, let me try to break this down step by step.First, I know that if a function passes through a point ( (x, y) ), then substituting ( x ) into the function should give me ( y ). So, since ( f(x) ) passes through ( (2, 1) ), I can plug in ( x = 2 ) and ( f(2) = 1 ) into the equation.So, substituting, I get:[1 = log_a(2 + b)]This equation relates ( a ) and ( b ). I can rewrite this logarithmic equation in exponential form to make it easier to handle. Remember, ( log_a(c) = d ) is equivalent to ( a^d = c ). Applying that here:[a^1 = 2 + b]Which simplifies to:[a = 2 + b]Okay, so that's one equation: ( a = 2 + b ). I'll keep that in mind.Now, moving on to the inverse function. The inverse function of ( f(x) ) is ( f^{-1}(x) ). The problem states that the graph of the inverse function passes through the point ( (2, 8) ). So, that means when ( x = 2 ), ( f^{-1}(2) = 8 ).But wait, I remember that if ( f^{-1}(c) = d ), then ( f(d) = c ). So, applying that here, since ( f^{-1}(2) = 8 ), it must be that ( f(8) = 2 ).So, substituting ( x = 8 ) into ( f(x) ), we get:[f(8) = log_a(8 + b) = 2]Again, I can rewrite this logarithmic equation in exponential form:[a^2 = 8 + b]So now I have another equation: ( a^2 = 8 + b ).Now, I have two equations:1. ( a = 2 + b )2. ( a^2 = 8 + b )I can use substitution to solve for ( a ) and ( b ). From the first equation, I can express ( b ) in terms of ( a ):[b = a - 2]Now, substitute this expression for ( b ) into the second equation:[a^2 = 8 + (a - 2)]Simplify the right side:[a^2 = 8 + a - 2]Which simplifies to:[a^2 = a + 6]Now, bring all terms to one side to form a quadratic equation:[a^2 - a - 6 = 0]I can try to factor this quadratic. Looking for two numbers that multiply to -6 and add to -1. Hmm, let's see: 2 and -3. Yes, because 2 * (-3) = -6 and 2 + (-3) = -1.So, factoring:[(a - 3)(a + 2) = 0]Setting each factor equal to zero gives:1. ( a - 3 = 0 ) => ( a = 3 )2. ( a + 2 = 0 ) => ( a = -2 )But wait, in the original function ( f(x) = log_a(x + b) ), the base ( a ) must satisfy ( a > 0 ) and ( a neq 1 ). So, ( a = -2 ) is invalid because it's negative. Therefore, the only valid solution is ( a = 3 ).Now, substitute ( a = 3 ) back into the first equation ( a = 2 + b ):[3 = 2 + b]Solving for ( b ):[b = 3 - 2 = 1]So, ( b = 1 ).Therefore, ( a = 3 ) and ( b = 1 ). The question asks for ( a + b ), so:[a + b = 3 + 1 = 4]Let me double-check my work to make sure I didn't make any mistakes.Starting with ( f(2) = 1 ):[log_3(2 + 1) = log_3(3) = 1]That's correct.Now, checking the inverse function. Since ( f(8) = 2 ), the inverse function should satisfy ( f^{-1}(2) = 8 ). Let me verify that.First, find the inverse function. To find ( f^{-1}(x) ), we start with:[y = log_3(x + 1)]Swap ( x ) and ( y ):[x = log_3(y + 1)]Now, solve for ( y ):[3^x = y + 1]So,[y = 3^x - 1]Therefore, ( f^{-1}(x) = 3^x - 1 ).Now, plugging ( x = 2 ) into the inverse function:[f^{-1}(2) = 3^2 - 1 = 9 - 1 = 8]Which matches the given point ( (2, 8) ). So, everything checks out.Thus, I'm confident that ( a = 3 ) and ( b = 1 ), so ( a + b = 4 ).**Final Answer**The value of ( a + b ) is boxed{4}.

answer：To determine the ordered pair (a, b), I need to find the values of a and b such that the cross product of the two vectors is the zero vector.First, I'll recall the formula for the cross product of two vectors in three-dimensional space. For vectors (mathbf{u} = begin{pmatrix} u_1 u_2 u_3 end{pmatrix}) and (mathbf{v} = begin{pmatrix} v_1 v_2 v_3 end{pmatrix}), the cross product (mathbf{u} times mathbf{v}) is given by:[mathbf{u} times mathbf{v} = begin{pmatrix} u_2v_3 - u_3v_2 u_3v_1 - u_1v_3 u_1v_2 - u_2v_1 end{pmatrix}]Applying this formula to the given vectors:[begin{pmatrix} 2 a -7 end{pmatrix} times begin{pmatrix} 5 4 b end{pmatrix} = begin{pmatrix} a cdot b - (-7) cdot 4 (-7) cdot 5 - 2 cdot b 2 cdot 4 - a cdot 5 end{pmatrix}]Simplifying each component:1. First component: (ab + 28)2. Second component: (-35 - 2b)3. Third component: (8 - 5a)Since the cross product equals the zero vector, each component must be zero:1. (ab + 28 = 0)2. (-35 - 2b = 0)3. (8 - 5a = 0)Solving the second equation for b:[-35 - 2b = 0 implies -2b = 35 implies b = -frac{35}{2}]Solving the third equation for a:[8 - 5a = 0 implies 5a = 8 implies a = frac{8}{5}]Finally, verifying these values in the first equation:[left(frac{8}{5}right) left(-frac{35}{2}right) + 28 = -28 + 28 = 0]Thus, the ordered pair is (left(frac{8}{5}, -frac{35}{2}right)).

answer：First, I need to understand the problem. The phrase "one and one-half of what number is 30" means that one and a half times an unknown number equals 30.To translate this into a mathematical equation, I can represent the unknown number as ( x ). Therefore, the equation becomes:[1.5 times x = 30]Next, I'll solve for ( x ) by dividing both sides of the equation by 1.5:[x = frac{30}{1.5}]Calculating the division gives:[x = 20]So, the unknown number is 20.