answer:To find the minimum distance from point O (the origin) to the line 2x - y + 1 = 0, we use the formula for finding the distance d from a point (x_0, y_0) to the line Ax + By + C = 0: d = frac { |Ax_0 + By_0 + C| }{ sqrt {A^2 + B^2} } Here, the coefficients from the line equation are A=2, B=-1, and C=1, and for point O, which is the origin, we have x_0=0 and y_0=0. Plugging these values into the distance formula gives: d = frac { |2(0) - 1(0) + 1| }{ sqrt {2^2 + (-1)^2} } = frac { |1| }{ sqrt {4 + 1} } = frac { 1 }{ sqrt {5} } = frac { sqrt {5} }{ 5 } Therefore, the minimum value of |OP| is boxed{frac { sqrt {5} }{ 5 }}.

answer:Let's denote the number of adult tickets sold as A and the number of senior citizen tickets sold as S. We are given the following two equations based on the information provided: 1) The total number of tickets sold is 510, so A + S = 510. 2) The total receipts were 8748 dollars, so 21A + 15S = 8748. We need to solve this system of equations to find the value of S, the number of senior citizen tickets sold. First, let's solve equation 1) for A: A = 510 - S Now, we can substitute this expression for A into equation 2): 21(510 - S) + 15S = 8748 Expanding the equation, we get: 10710 - 21S + 15S = 8748 Combining like terms, we get: -6S = 8748 - 10710 -6S = -1962 Now, divide both sides by -6 to solve for S: S = -1962 / -6 S = 327 So, boxed{327} senior citizen tickets were sold.

answer:Given the formula for the length of a metal rod as a function of temperature: [ l = l_{0}(1 + alpha t) ] where: - ( l_0 ) denotes the length of the metal rod at ( t = 0 ) (initial temperature), - ( t ) is the temperature, - ( alpha ) is a constant known as the coefficient of linear expansion. Let us determine how ( l ) varies as a function of ( t ) and understand what the coefficient ( alpha ) represents. 1. **Expression Analysis**: - The formula ( l = l_0(1 + alpha t) ) is a linear function of the temperature ( t ) since it can be rewritten as: [ l = l_0 + l_0 alpha t ] This shows that ( l ) increases linearly with ( t ). 2. **Function Type**: - From the expression, it is clear that ( l(t) = l_0 + l_0 alpha t ) represents a linear function of ( t ). Hence, the length ( l ) is a linear function of the temperature ( t ). 3. **Determining ( alpha )**: - The coefficient ( alpha ) is known as the coefficient of linear expansion. It indicates how much the length of the rod changes per unit temperature change. - Specifically, ( alpha ) is defined as the fractional change in length per degree of temperature change. This can be interpreted as the change in length per degree Celsius. - Mathematically, if the length ( l ) changes from ( l_0 ) to ( l ) when the temperature increases from ( 0^circ C ) to ( 1^circ C ), then: [ Delta l = l_0 alpha ] where ( Delta l ) is the increase in length over 1 degree Celsius. **Conclusion**: - The length ( l ) is a linear function of the temperature ( t ). - The constant ( alpha ) represents the increase in length per 1 degree Celsius rise in temperature. [ boxed{t text{ is a linear function of } l; alpha text{ is the increase in length of the rod in cm when the temperature rises from } 0^circ text{ to } 1^circ text{C}} ]

answer:Let ( x ) be the number of words I learn. The total number of words is 800. From the words I did not learn, 10% are guessed correctly. First, calculate the total number of words I get correct, which includes words learned and words guessed correctly: [ text{Words learned correctly} = x ] [ text{Words guessed correctly} = 0.10 times (800 - x) ] Setting up the equation to ensure a total of 90% correct answers: [ x + 0.10 times (800 - x) = 0.90 times 800 ] Simplify and solve for ( x ): [ x + 80 - 0.10x = 720 ] [ 0.90x = 640 ] [ x = frac{640}{0.90} approx 711.111 ] Since ( x ) must be a whole number, round up to ensure reaching at least 90%: [ x = boxed{712} ] Conclusion: Learning at least 712 words ensures I get at least 90% on the exam, considering that I can correctly guess 10% of the unlearned words.