5 7 $$ \left(x^{2} y-3 y^{2}+5 x y^{2}\right)-\left(-x^{2} y+3 x y^{2}-3 y^{2}\right) $$ Which of the following is equivalent to the expression above? A) $$ 4 x^{2} y^{2} $$ B) $$ 8 x y^{2}-6 y^{2} $$ C) $$ 2 x^{2} y+2 x y^{2} $$ D) $$ 2 x^{2} y+8 x y^{2}-6 y^{2} $$ 6 $$ h=3 a+28.6 $$ A pediatrician uses the model above to estimate the height $$ h $$ of a boy, in inches, in terms of the boy's age $$ a $$, in years, between the ages of 2 and 5 . Based on the model, what is the estimated increase, in inches, of a boy's height each year? A) 3 B) 5.7 C) 9.5 D) 14.3 $$ m=\frac{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}{\left(1+\frac{r}{1,200}\right)^{N}-1} P $$ The formula above gives the monthly payment $$ m $$ needed to pay off a loan of $$ P $$ dollars at $$ r $$ percent annual interest over $$ N $$ months. Which of the following gives $$ P $$ in terms of $$ m, r $$, and $$ N $$ ? A) $$ P=\frac{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}{\left(1+\frac{r}{1,200}\right)^{N}-1} m $$ B) $$ P=\frac{\left(1+\frac{r}{1,200}\right)^{N}-1}{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}} m $$ C) $$ P=\left(\frac{r}{1,200}\right) m $$ D) $$ P=\left(\frac{1,200}{r}\right) m $$

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UpStudy AI · Accepted Answer

1) Simplify 
$$
\left(x^{2} y-3 y^{2}+5 x y^{2}\right)-\left(-x^{2} y+3 x y^{2}-3 y^{2}\right)
$$
 • Distribute the minus sign:
$$
x^{2} y-3 y^{2}+5xy^{2}+x^{2} y-3xy^{2}+3y^{2}
$$
 • Combine like terms:
  – $$x^{2} y+x^{2} y=2x^{2}y$$
  – $$-3 y^{2}+3 y^{2}=0$$
  – $$5xy^{2}-3xy^{2}=2xy^{2}$$
 • The simplified expression is:
$$
2x^{2}y+2xy^{2}
$$
Thus, the answer is Option C: $$2x^{2} y+2xy^{2}$$.

2) The model for a boy’s height is:
$$
h=3a+28.6.
$$
 • The coefficient of $$a$$ is 3.
 • This indicates that the height increases by 3 inches per year.
Thus, the answer is Option A: 3.

3) We are given the monthly payment formula
$$
m=\frac{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}{\left(1+\frac{r}{1,200}\right)^{N}-1}\,P.
$$
Solve for $$P$$:
 • Multiply both sides by the denominator:
$$
m\left[\left(1+\frac{r}{1,200}\right)^{N}-1\right]=\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}P.
$$
 • Isolate $$P$$ by dividing both sides by $$\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}$$:
$$
P=\frac{\left[\left(1+\frac{r}{1,200}\right)^{N}-1\right]}{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}\,m.
$$
Thus, the answer is Option B:
$$
P=\frac{\left(1+\frac{r}{1,200}\right)^{N}-1}{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}\,m.
$$
1. The simplified expression is $$2x^{2}y + 2xy^{2}$$. 2. The estimated increase in height each year is 3 inches. 3. The formula for $$P$$ in terms of $$m$$, $$r$$, and $$N$$ is: $$ P=\frac{\left(1+\frac{r}{1,200}\right)^{N}-1}{\left(\frac{r}{1,200}\right)\left(1+\frac{r}{1,200}\right)^{N}}\,m. $$

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