Use a variation model to solve for the unknown value. The...

Use a variation model to solve for the unknown value. The amount of a pain reliever that a physician prescribes for a child varies directly as the weight of the child. A physician prescribes $180 \mathrm{mg}$ of the medicine for a 40-lb child. (See Example 3$)$ a. How much medicine would be prescribed for a 50-lb child? b. How much would be prescribed for a $60-1 \mathrm{~b}$ child? c. How much would be prescribed for a 70 -lb child? d. If $135 \mathrm{mg}$ of medicine is prescribed, what is the weight of the child?

Use a variation model to solve for the unknown value.
The amount of a pain reliever that a physician prescribes for a child varies directly as the weight of the child. A physician prescribes $180 \mathrm{mg}$ of the medicine for a 40-lb child. (See Example 3$)$
a. How much medicine would be prescribed for a 50-lb child?
b. How much would be prescribed for a $60-1 \mathrm{~b}$ child?
c. How much would be prescribed for a 70 -lb child?
d. If $135 \mathrm{mg}$ of medicine is prescribed, what is the weight of the child?

Key Concepts

Direct Variation

Direct variation describes a relationship between two variables in which one variable is a constant multiple of the other. This means that as one variable increases, the other increases in a proportional manner, and the ratio between the two variables remains constant.

Constant of Proportionality

The constant of proportionality is the fixed number that links the two variables in a direct variation relationship. It is found by dividing one of the known pairs of variables and remains the same for all pairs, allowing the formulation of the equation where one quantity equals the constant multiplied by the other.

Setting Up Proportional Equations

When given a scenario where quantities vary directly, a proportional equation of the form y = kx is used. This equation is set up by identifying one variable as dependent on the other, finding the constant of proportionality with one pair of values, and then applying it to compute unknown values for other pairs.

Solving Real-World Problems with Proportions

In practical applications, such as determining dosages or scaling recipes, understanding and using proportion models is crucial. Once the direct variation relationship and constant of proportionality are established, the model can be used to solve for unknowns by setting up equations or proportions that equate the known ratios to the unknown variables.

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