Question

According to the Ehrenberg relation, the average measurements of heights, \(h,\) in centimetres, and masses, \(m,\) in kilograms, of children between the ages of 5 and 13 are related by the function \(\log m=0.008 h+0.4\) a) Predict the height of a 10-year-old child with a mass of \(60 \mathrm{kg},\) to the nearest centimetre. b) Predict the mass of a 12 -year-old child with a height of \(150 \mathrm{cm},\) to the nearest kilogram.

Short answer

a) Height is approximately 172 cm. b) Mass is approximately 40 kg.

Step-by-step solution

Identify the given values for part (a)

For part (a), the mass of the child is given as 60 kg. We need to find the height, denoted as h.

Substitute the given mass into the equation for part (a)

Use the equation \ \ \( \ \ \log m = 0.008 h +0.4 \). Substitute \(m = 60\) into the equation: \ \ \( \ \ \log(60)=0.008 h+0.4 \)

Isolate the height variable (h)

Subtract 0.4 from both sides to isolate the term with height: \ \ \( \ \ \log(60)-0.4=0.008h \)

Calculate the right side of the equation

Use a calculator to compute the value of \( \log(60)\): \ \ \( \ \ \log(60) \approx 1.7782 \). Thus, \ \ \( \ \ 1.7782 - 0.4 \approx 1.3782 \)

Solve for height (h)

Divide both sides by 0.008 to find \(h\): \ \ \( \ \ h = \frac{1.3782}{0.008} \approx 172.275 \). Rounded to the nearest centimeter: \(h \approx 172\;\text{cm}\)

Identify the given values for part (b)

For part (b), the height of the child is given as 150 cm. We need to find the mass, denoted as m.

Substitute the given height into the equation for part (b)

Use the equation \ \ \( \ \ \log m = 0.008 h +0.4 \). Substitute \(h = 150\) into the equation: \ \ \( \ \ \log m = 0.008 \times 150 + 0.4 \)

Compute the value inside the logarithm

Calculate the multiplication: \ \ \( \ \ 0.008 \times 150 = 1.2 \). Then add 0.4: \ \ \( \ \ \log m = 1.2 + 0.4 = 1.6 \)

Solve for mass (m)

Rewrite the equation to solve for \(m\): \ \ \( \ \ m = 10^{1.6} \). Use a calculator to find \(10^{1.6} \approx 39.81\). Rounded to the nearest kilogram: \(m \approx 40\;\text{kg}\)

Key concepts

Logarithmic Functions

Logarithmic functions are very helpful in mathematics, especially when dealing with growth-related problems. A logarithmic function is the inverse of an exponential function. In simpler terms, if you think about exponential growth, which multiplies, logarithmic functions can help undo that multiplication by 'shrinking' the numbers back down. For instance, in the problem we’re solving, the function \(\log m = 0.008 h + 0.4\) lets us find relationships between height (\( h \)) and mass (\( m \)). This specific logarithmic function tells us how the mass of a child grows relative to their height.

Algebraic Manipulation

Algebraic manipulation involves rearranging and solving equations, which is a crucial skill in mathematics. For part (a) of the exercise, we start by substituting the given mass (\( m = 60 \) kg) into the function \(\log(60) = 0.008h + 0.4\). After substituting, we need to isolate the variable \( h \), which represents the height. By subtracting 0.4 from both sides of the equation, we isolate the term with \( h \): \(\log(60) - 0.4 = 0.008h\). Lastly, we divide by 0.008 to solve for \( h \). This process shows how algebraic manipulation can uncover information hidden in mathematical functions.

Growth Prediction

Growth prediction is a key theme in problems involving biological data such as height and weight. Using the Ehrenberg relation helps us make predictions about child growth patterns. For example, in part (a), we use the given mass to predict height. In part (b), we use a given height to estimate mass. The function \(\log m = 0.008 h + 0.4\) embodies how we can mathematically predict growth trends. These predictions can be vital for various applications like pediatric healthcare, where understanding such trends can help in monitoring child development.

Child Height and Weight Correlation

The correlation between child height and weight is an important aspect of child growth studies. The Ehrenberg relation shows a mathematical way to represent this correlation. From the equation \(\log m = 0.008 h + 0.4\), we see that as the height (\( h \)) increases, the mass (\( m \)) also increases. This functional relationship helps visualize how a change in height could influence weight. Such correlations are very valuable, as they provide healthcare professionals and researchers a way to predict one measurement given the other, thus aiding in better understanding of child growth patterns.