If A is a 2 x 3 matrix and B is a 3 x 4 matrix, how many columns does AB have? Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their f... If A is a 2 x 3 matrix and B is a 3 x 4 matrix, how many columns does AB have? Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, what is the distance between their tops? If triangle ABC is an isosceles triangle with ∠C=90° and AC=5cm, then AB is? The straight line given by the equation x=11 is parallel to which axis? If two non-vertical lines are perpendicular if and only if m1=m2, m1=m2, m1m2=-1? If the ratio of the height of a tower and the length of its shadow is √3:1, then the angle of elevation of the sun has measure? Read more

Steps to Solve

1. Understanding Matrix Multiplication To find the number of columns in the matrix product $AB$, we need to know the dimensions of both matrices. Matrix $A$ is $2 \times 3$ (2 rows and 3 columns) and matrix $B$ is $3 \times 4$ (3 rows and 4 columns).

2. Determining the Resulting Matrix Dimensions The number of columns in the resulting matrix $AB$ equals the number of columns in matrix $B$. Therefore, $AB$ will have:

   • Rows from $A$: 2
   • Columns from $B$: 4

3. Finding the Distance Between the Tops of Poles Using the Pythagorean theorem for two poles of heights 6 m and 11 m and base distance of 12 m, the height difference is: $$ h = 11 - 6 = 5 \text{ m} $$ The distance between the tops is calculated as: $$ d = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ m} $$

4. Finding the Length of Side AB in Triangle ABC Given an isosceles triangle where $C = 90^\circ$, $AC = 5 \text{ cm}$, we apply Pythagorean theorem: $$ AB = \sqrt{AC^2 + BC^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \text{ cm} $$

5. Identifying Lines from Their Equation The equation $X = 11$ represents a vertical line. Therefore, it is:

   • Parallel to Y-axis.

6. Checking the Condition for Perpendicular Lines For two non-vertical lines to be perpendicular, the product of their slopes must equal $-1$, i.e., $m_1m_2 = -1$.

7. Calculating the Angle of Elevation of the Sun Given the ratio of height $h$ of the tower to the length of shadow $l$ as $\frac{h}{l} = \sqrt{3}$, we use the tangent for elevation angle: $$ \tan(\theta) = \frac{h}{l} \Rightarrow \theta = 60^\circ $$