Question

in triangle xyz, angle y is a right - angle. point p lies on xz, and point q lies on yz such that pq is parallel to xy. if the measure of angle xzy is 63°, what is the measure, in degrees, of angle xpq?
an investment account was opened with an initial value of $890. the value of the account doubled every 10 years. which equation represents the value of the account m(t), in dollars, t years after the account was opened?
a) m(t)=890(\frac{1}{2})^{\frac{t}{10}}
b) m(t)=890(\frac{1}{10})^{\frac{t}{1}}
c) m(t)=890(2)^{\frac{t}{10}}
d) m(t)=890(10)^{\frac{t}{1}}

Explanation:

Step1: Analyze parallel - line properties

Since $PQ\parallel XY$ and $\angle Y = 90^{\circ}$, in right - triangle $XYZ$ with $\angle XZY=63^{\circ}$, and considering the relationship between parallel lines. The angle $\angle XPQ$ and $\angle XYZ$ are corresponding angles with respect to the parallel lines $PQ$ and $XY$ and the transversal $XZ$.

Step2: Use angle - sum property of a triangle

In $\triangle XYZ$, we know that the sum of the interior angles of a triangle is $180^{\circ}$. But we can also use the fact that for parallel lines $PQ$ and $XY$, $\angle XPQ$ and $\angle XYZ$ are related. Since $PQ\parallel XY$, $\angle XPQ = 180^{\circ}-\angle PQY$. And in $\triangle PQY$, $\angle PQY = 90^{\circ}$ (because $\angle Y = 90^{\circ}$ and $PQ\parallel XY$). Also, we can consider the fact that the alternate - interior angle relationship. The angle $\angle XPQ$ and the angle formed by the non - parallel sides with respect to the parallel lines has a specific relationship. Since $PQ\parallel XY$, $\angle XPQ=153^{\circ}$. We know that $\angle XZY = 63^{\circ}$, and the angle adjacent to $\angle XPQ$ in the non - right triangle formed by $PQ$ and $XZ$ is $27^{\circ}$ (because in $\triangle XYZ$, the third angle $\angle YXZ=180^{\circ}-90^{\circ}-63^{\circ}=27^{\circ}$). And $\angle XPQ = 180^{\circ}-27^{\circ}=153^{\circ}$.

for the second part:

Step1: Identify the compound - growth formula

The general formula for compound growth is $A = P(1 + r)^n$, where $P$ is the initial amount, $r$ is the growth rate per period, and $n$ is the number of periods. In the case of an investment that doubles every $10$ years, the initial value $P = 890$. The growth factor is $2$ (since the value doubles), and the number of $10$ - year periods in $t$ years is $\frac{t}{10}$.

Step2: Write the function

Substituting the values into the formula, we get $M(t)=890\times2^{\frac{t}{10}}$.

Answer:

$153$