Question

1. in rectangle abcd shown, segments \\( \overline { b e } \\) and \\( \overline { c e } \\) partition the rectangle into 3 triangles. given \\( d e = 7 \\) centimeters, \\( b e = 26 \\) centimeters, and \\( c e = 25 \\) centimeters, what is the length, in centimeters, of \\( \overline { b c } \\)?

a. 10
b. 15
c. 17
d. 24

1. in a particular cleaning solution, the ratio of concentrated solution to water is \\( 3 : 40 \\). how many cups of concentrated solution should be added to 5 gallons of water to make the cleaning solution in the given ratio?

(note: \\( 4 \\) cups \\( = 1 \\) quart; \\( 4 \\) quarts \\( = 1 \\) gallon)
f. 12
g. 6
h. \\( 1 \frac { 1 } { 2 } \\)
j. \\( \frac { 3 } { 8 } \\)

1. let \\( f ( t ) = 7 e ^ { 3 t } + 1 \\). which of the following numbers is closest to the value of \\( f ( 5 ) \\)?

a. \\( - 2 \times 10 ^ { - 1 } \\)
b. \\( 3 \times 10 ^ { 2 } \\)
c. \\( 2 \times 10 ^ { 7 } \\)
d. \\( 6 \times 10 ^ { 7 } \\)

1. which of the following expresses \\( 40 ^ { \circ } \\) in radians?

f. \\( \frac { 2 } { 9 \pi } \\)
g. \\( \frac { 2 \pi } { 9 } \\)
h. \\( \frac { 9 \pi } { 2 } \\)
j. \\( \frac { 7,200 } { \pi } \\)

Explanation:

Step1: Use the Pythagorean theorem in right - triangle \(DEC\)

Let \(DC = x\). In right - triangle \(DEC\), by the Pythagorean theorem \(DC^{2}+DE^{2}=CE^{2}\). Substitute \(DE = 7\) and \(CE = 25\) into the formula: \(x^{2}+7^{2}=25^{2}\). Then \(x^{2}=25^{2}-7^{2}=(25 + 7)(25 - 7)\) (using the difference of squares \(a^{2}-b^{2}=(a + b)(a - b)\)). So \(x^{2}=(32)(18)=576\), and \(x = 24\).

Step2: Use the Pythagorean theorem in right - triangle \(ABE\)

Since \(ABCD\) is a rectangle, \(AB = DC=24\). In right - triangle \(ABE\), let \(AE = y\). By the Pythagorean theorem \(y^{2}+24^{2}=26^{2}\). Then \(y^{2}=26^{2}-24^{2}=(26 + 24)(26 - 24)=(50)(2)=100\), and \(y = 10\).

Step3: Calculate the length of \(AD\) (and \(BC\) since \(AD = BC\) in a rectangle)

\(AD=AE + DE\). Substitute \(AE = 10\) and \(DE = 7\) into the formula. So \(AD=10 + 7=17\).

Answer:

C. 17