At a construction site, to transport the earth extracted for the construction of a building's foundations, three different types of trucks are used: A, B, and C. Type A trucks have a capacity of 14 tonnes, type B trucks have 24 tonnes, and type C trucks have 28 tonnes. One more type A truck would be needed to equal the number of remaining trucks. 10\% of the capacity of all type B trucks equals one-seventh of the capacity of the largest tonnage trucks. Today, with each truck making a single trip at maximum capacity, 302 tonnes of earth have been extracted from the site. How much earth has been transported today by trucks of each type?
Given the function $f ( x ) = \sqrt [ 3 ] { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$, find:\ a) ( 0.25 points) Study whether it is even or odd.\ b) ( 0.75 points) Study its differentiability at the point $x = 1$.\ c) (1.5 points) Study its relative and absolute extrema.
Given the points $\mathrm { A } ( 1 , - 2,3 ) , \mathrm { B } ( 0,2 , - 1 )$ and $\mathrm { C } ( 2,1,0 )$. Find:\ a) (1.25 points) Verify that they form a triangle $T$ and find an equation of the plane containing them.\ b) ( 0.75 points) Calculate the intersection of the line passing through points A and B with the plane $z = 1$.\ c) ( 0.5 points) Determine the perimeter of triangle T.
An event A has probability $\mathrm { P } ( \mathrm { A } ) = 0.3$.\ a) ( 0.75 points) An event B with probability $\mathrm { P } ( \mathrm { B } ) = 0.5$ is independent of A. Calculate $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } )$.\ b) ( 0.75 points) Another event C satisfies $P ( C \mid A ) = 0.5$. Determine $P ( A \cap \bar { C } )$.\ c) (1 point) If there is an event D such that $P ( \bar { A } \mid D ) = 0.2$ and $P ( D \mid A ) = 0.5$, calculate $\mathrm { P } ( \mathrm { D } )$.
Given the system $\left\{ \begin{array} { c } ( a + 1 ) x + 4 y = 0 \\ ( a - 1 ) y + z = 3 \\ 4 x + 2 a y + z = 3 \end{array} \right.$, find:\ a) ( 1.25 points) Discuss it as a function of the parameter $a$.\ b) ( 0.5 points) Solve it for $a = 3$.\ c) (0.75 points) Solve it for $a = 5$.
Given the real function of a real variable defined on its domain as $f ( x ) = \left\{ \begin{array} { l l l } \frac { x ^ { 2 } } { 2 + x ^ { 2 } } & \text { if } & x \leq - 1 \\ \frac { 2 x ^ { 2 } } { 3 - 3 x } & \text { if } & x > - 1 \end{array} \right.$, find:\ a) ( 0.75 points) Study the continuity of the function on $\mathbb{R}$.\ b) (1 point) Calculate the following limit: $\lim _ { x \rightarrow - \infty } f ( x ) ^ { 2 x ^ { 2 } - 1 }$.\ c) (0.75 points) Calculate the following integral: $\int _ { - 1 } ^ { 0 } f ( x ) d x$.
Given the line $r \equiv \frac { x - 1 } { 2 } = \frac { y } { 1 } = \frac { z + 1 } { - 2 }$, the plane $\pi : x - z = 2$ and the point A(1,1,1), find:\ a) ( 0.75 points) Study the relative position of $r$ and $\pi$ and calculate their intersection, if it exists.\ b) ( 0.75 points) Calculate the orthogonal projection of point A onto the plane $\pi$.\ c) (1 point) Calculate the point symmetric to point A with respect to the line $r$.
The length of the Pacific sardine (Sardinops sagax) can be considered a random variable with normal distribution with mean 175 mm and standard deviation 25.75 mm.\ a) (1 point) A canning company for this variety of sardines only accepts as quality sardines those with a length greater than 16 cm. What percentage of the sardines caught by a fishing vessel will be of the quality expected by the canning company?\ b) ( 0.5 points) Find a length t $< 175 \mathrm {~mm}$ such that between t and 175 mm there are 18\% of the sardines caught.\ c) (1 point) At sea, sardines are processed in batches of 10. Subsequently, sardines from each batch that are smaller than 15 cm are returned to the sea as they are considered small. What is the probability that in a batch there is at least one sardine returned as small?