Question

How high is the building with a shadow \(19 \mathrm{~m}\) long? (1) We look up on the internet the angle of the sun for that time and place (2) At the same time, a gatepost of known height casts a shadow \(1.5 \mathrm{~m}\) long A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

B. 2 alone, not 1 alone

Step-by-step solution

Understanding the Problem

We need to find the height of a building given the length of its shadow and additional data involving a gatepost and the angle of the sun.

Identify the Given Information

The building's shadow length is 19 meters. A gatepost of known height casts a shadow 1.5 meters long. There is also the angle of the sun.

Consider How to Use Information 1

Look up the angle of the sun. Using trigonometric relationships, we can find the height if we know the angle.

Evaluate Information 1

Information 1 alone may not be sufficient because while it gives the sun angle, it does not directly provide the relationship needed to find the building height without an additional reference.

Consider How to Use Information 2

A gatepost of known height and its shadow length. By knowing one shadow length and height, we can use proportions to find the building height.

Evaluate Information 2

Information 2 alone provides a relationship between known height and shadow length that can be applied to find the building height.

Final Evaluation

Information 2 alone (using the proportion between the gatepost height and shadow length) is sufficient.

Key concepts

trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In this exercise, the relationship between the height of the building and the length of its shadow is key. By knowing the angle of elevation of the sun (which can be looked up online), we use trigonometric ratios such as the tangent function. For a right-angle triangle, the tangent of the angle is the ratio of the opposite side (height of the building) to the adjacent side (length of the shadow). Therefore, if the angle \( \theta \) and the length of the shadow \( b \) are known, the height \( h \) of the building can be determined using the formula:
\[ \tan{\theta} = \frac{h}{b} \]

From this, rearranging gives the height as: \[ h = b \times \tan{\theta} \]

However, without knowing the value of \( h \) (building height) or more specific details, simply knowing the length of the shadow is insufficient. Hence, the trigonometric information must be paired with another piece of information for a complete solution.

proportional reasoning

Proportional reasoning is the ability to make comparisons and understand the equivalence between ratios. In the given problem, we compare the shadows of two different objects. The known height of a gatepost and its shadow length gives us a proportion that can be applied. For instance, if a gatepost that is \( h_g \) tall casts a shadow of length \( s_g \), and a building casts a shadow of \( s_b \), we can set up a proportion to find the building’s height \( h_b \):

\[ \frac{h_g}{s_g} = \frac{h_b}{s_b} \]

Rearranging, the formula to find the building’s height is:
\[ h_b = \frac{h_g \times s_b}{s_g} \]

By using the known values from the gatepost, we directly relate them to the building, making it possible to find the height with this approach. Hence, in this specific exercise, the proportional reasoning method provides a direct and adequate solution.

graduate management admission test

The Graduate Management Admission Test (GMAT) assesses various capabilities, including quantitative reasoning, critical thinking, and problem-solving. This problem involves skills in both trigonometry and proportional reasoning, crucial areas under the GMAT quantitative section. Candidates are often tested on their ability to apply mathematical concepts to solve real-world problems efficiently.

The ability to use given data points, pick useful information, and apply correct reasoning methods is essential for success. This exercise exemplifies how critical it is to determine which pieces of information are sufficient for solving a problem. It also highlights the importance of flexibility in employing different methods, be it trigonometric ratios or proportional reasoning, to arrive at the solution accurately.

Practicing such problems helps improve not only test-taking speed but also enhances deeper understanding and application of mathematical principles, both necessary for achieving a high GMAT score.